e infty


1 'Edited by Dold and B. Eckmann A J. May Peter - E, Ring Spaces and E, Ring Spectra with contributions by Frank Quinn, Nigel Ray, Tornehave Jnrrgen and Springer-Verlag Berlin Heidelberg New York

2 Contents AMS Classifications (1970): 18D10, 18F25, 55B15, 55820, Subject 55D35,55E50,55F15,55F25,55F35,55F45,55F50,55F60,55625 ISBN Springer-Verlag Berlin . Heidelberg . New York 3-540-08136-4 ISBN Springer-Verlag New York . Heidelberg . Berlin 0-387-08136-4 the whole work to copyright. All rights are resewed, whether is subject This or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, in data banks. and storage Under 5 54 German Copyright Law where copies are made for other of the than private use, a fee is payable to the publisher, the amount of the fee to be determined agreement with the publisher. by O by Springer-Verlag Berlin . Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, HemsbachlBergstr. 214113140-543210

3 ... 139 ring spaces and bipermutative categories VI. Chapter Ew ... 1. of E ring spaces 139 5 The definition w - ... 146 examples of operad pairs $ 2. Units; ' i , 3 there In topology, classes ... general two is a dichotomy between 93. Symmetric bipermutative categories bimonoidal and 151 thinking On the one @paces and ways of hand, roles. about their ... 157 Bipermutative and categories spaces. E w ring $4. are concrete spaces, most importantly the various geometric are the ... bipermutative of Examples 5. 5 160 categories problems one Typical manifolds. of pas about study to proposes ... 168 principle for recognition The E spaces ring w VII. Chapter cobordism, least h spaces are their classification, at up to and ... 169 convex bodies operads little The 1. $ to the existence of an equivalence of a given space obstructions ... 176 The canonical E w operad pair 5 2. and fibrations h u space with a richer type of structure. Bundles ... one The 3. 5 recognition principle 180 operad a the spaces generally play geometric central role in solution r ... problems. On the other hand, there spaces are the classifying ouch principle The $4. operad 188 two recognition and invariants bundle and fibration theories other cohomological ... $5. The multiplicative Ew structure and localizations. 194.- are These of paces. geo- analysis the for tools as of thought ... 201 Chapter VIII. Algebraic and topological K-theory. on the fact that theorems it is and c problems, a familiar ... $ 1. Examples; algebraic K-theory 203 to translate level often space #&iifying intrinsic informa- yield ... 2. 212 5 Bott periodicity and Brauer lifting bundle example, n periodicity Bott on for Thus, level. theory the ... fields, Frobenius, 3 and 223 Finite 3. $ B Goker a as inated classifying of types homotopy the about stateme2t ... i primes 235 odd SF at of splitting The $4. but usefully interpreted as a statement about bundles u, is most I ... tensor products. 244 theory space loop infinite in Pairings IX. Chapter homotopy the of intensive analysis an seen has decade last ... Q-spaces of and categories of Pairings 1. 5 246 of spaces classifying view the a with geometric topology, ... 250 for recognition principle The 2. $ pairings and obstruction theory pplications to the classification ... 257 Bibliography differentiable manifolds. less richly structured than 11 ... !I Index. 263 very much structure than the mere homotopy type more is The of topological and PL manifolds and of Poincare study spaces forces consideration of bundle and fibration theories ity ool~ornology of whose classifying spaces is wholly inaccessible ho techniaues and invariants of homotopy theory. classical The appropriate framework for the study of these classifying spaces is lriCo loop space theory and, in particular, its multiplicative elaboration

4 I' / ' (, ,[.. 2 3 an The prototype of sphere spectrum Em %so. the is ring spectrum also This for the appropriate framework is which of this is book the theme development of K-theory, by which we understand the most structured algebraic the of categories, and one of theory main goals of this homotopy the discrete structure space loop additive infinite its ip and the multiplicative between classifying spaces between the relationship of analysis a complete is volume the the SF identity its of component element. on structure lnitc loop space is It spaces topology algebraic geometric of K-theory. loop infinite and the of The ' win IIC~ hterrelationships which be codified in our basic definitions. are presented have powerful consequences, obtained calculational results which 0 ignore If we QS on and concentrate on SF, then the interest focuses For they example, make it possible to pass quite directly from in [26]. relationships topology in particular, on c~rnctric and, among F, Top, PL, the finite representation theoretical of the homologies of appropriate computations classifying clnssi'cal groups, and their homogeneous and spaces. We give topological bundles of to detailed characteristic classes for groups analysis 2).and spherical fibrations. from (away in later recapitulated (and developed [45] in this spaces Em of tlloory space of infinite topology, algebraic classical view of point the From loop be thought of as of unstable methods for the study of stable theory may the use theory. point is a recognition principle for infinite loop homotopy Its starting that allows one to pass back and forth spaces spectra and spaces with between de- of Our infinite loop space structure on SF construction an structure, namely E spaces. We shall enrich this appropriate internal m on use not just of the SF(n) but of the SFW) for all finite- If this see to are we flional product spaces inner real VCR-. additive allows one to pass back theory structure which with a multiplicative between and forth and E ring E spectra ring spaces. w w theory is different in kind from the old: the Conceptually, the new chapter in npectra mainly an and summary 11,which extract is structure on appropriate is itself unstable in that it multiplicative spectra of that slight amount of information about coordinate- 1481 book. Bpectra and th~ stable homotopy category needed in this not an equivalent formulation expressible solely in terms of admit to appears definition which IV, rele- WQ reach the chapter of E, ring spectra in is the because This the stable homotopy category. structure visible to Int work with Frank Quinn and Nigel Ray. The fundamental idea level on vant structure requires very precise algebraic data on the point-set tting an space structure on a spectrum in order to obtain Em that The applications the spectrum. spaces comprise together the which is entirely due to spectrum chest possible notion of a ring concern us center around space level exploitation of this algebraic data. here Quinn. manageable and technically correct way of doing this A It has very recently become possible to express a significant portion of the leading us time. The essential insight some to the correct for This reformulation leads good that out pointed who concrete tion came from Nigel Ray, category. of terms structure in stable maps in the uctions of Thom spectra gave naturally occuring examples of the subject of a future to applications in stable homotopy theory and will be volume.

5 !, , ...,' .'..,) ,,~ .. .) , - .I > \.,'/ ,- ! \- !. . .,,) L- ,) , \.-, 1 . . , "\ ,I . \-., 1 \,,.I \ .,I / >.~- >,.,,! ; 1 , ,! '. I \ , , ...-) ...) 1 \.., $:- 1 .-) '<-I ,. ~, i ,: ,,-.; ", ,I . \b.) 4 more eo. Much significantly, it encodes the interrelationship and that these examples with of structure kind the right spectra by determLned BO spectrum WQcn kO and the . ~t is one easy definition. taken as models the general on which could be t~'~attern (3 interrelationship aocluence of this which is exploited in chapter family One key topology of classifying is spaces in geometric from chapter I, namely the classifying spaces for sphere missing Em therefore in nature as an "r ring spectrum and must be man- of a given with respect to bundles a specified type oriented cohomo- classifying In fact, no concrete constructions of such logy theory. We remedy this in chapter I11 by literature. exist in the sDaces This brings us to the which, aside from last four chapters for fibrations and bundles use of the general classification theory in the first sections I, of definitions contained of chapters In 53 of chapter IV, we give these classifying developed in [471. In chapter and independent of the first largely IV,are five. spaces infinite loop space structures when the specified cohomology wa define Em ring spaces and show that the classifying spaces I Em ring spectrum. represented by an theory is bipermuta- appropriate structure, namely with internal ntegories of such In chapter spaces. categories, are examples we VII, on the class-' V, we demonstrate In chapter that formal analysis the machine for the manufacture 461 el constructed in [45 and one to deduce allows ifying space level of the sharpened versions out coordinate-free it turn to make spaces so as Em from poctra on topological and of Sullivan on [4,5] of Adams results [721 J(X) We then ha. show that if the machine is fed the additive (away 2) from the k0-orientation of from the prime bundle theory Em out an of an it turns then Lure ring spectrum. Em ring space, together of ko[l/2]-orientation the and Spin-bundles STop-bundles mmediate application is a multiplicatively enriched version from operations, Adams the derived v,ith the cannibalistic classes Q~Q, effect that to the [16,68,46] theorem Barratt-Quillen 10 orientations, specified by use of the them and the conjecture. dams Chua the stable homotopy groups of spheres, can be constructed throughout this we section, combine various last from the results infinite loop Our version shows that the symmetric groups. of and Priddy volume Snaithl and Adams of results with [a], Madsen, SF, and thus 0 the classifying space BSF for stable spherical [42], to analyze the infinite loop space and Ligaard Tornehave I381 tions, can also be constructed out of symmetric groups. (away from 2) and of various other classifying BTop structure of the last section of chapter VII, we study the spectra turned In his material spaces utilized in earlier parts of the chapter. whon the machine is fed the multiplicative structure of an E~ completes most in a preprint version of bf the program envisioned In particular, we obtain a multiplicative version apace. purely this chapter. relationship between SF and symmetric groups. VIII, In chapter largely independent section, last from chapter V is Aside its o the promised analysis of the relationship between the classifying self-contained- of infinite loop space theory and is reasonably of geometric topology and the infinite loop spaces of algebraic fact However, its earlier sections essential use of the do make ry. This basic material is a mosaic of'results due to Jfirgen Tornehave spectrum that real connective K-theory is represented by an Em ring and generalizations oolf and includes new proofs of the results the component of implies in particular that BO fact ko. This 8' nlly given in his thesis [75] and in his unpublished preprints is an infinite loop the identity element of the zeroth space of k0, - \

6 [76] The authorship. here under joint it is presented ; [77] and materials -- us to fit together different box which allow of our black algebrilic K-theory was established and topological connection between theory. parts of the giver. by Quillen that and we show [58,591, the maps in the work of with a discussion of how We illustrate this point BO ap2ears in our are conjecture the to prove he used which lifting Brauer ~'.darr,s theory. in chapter I, the ordinary classifying space As explained Via the and multiplicatively. loop maps, both additively infinite an infinite and thus space Em of the orthogonal group is an infinite understanding good this information yields a Probenius automorphism, loop space. VI, As explained in chapter Em is an ring BO(n) n of the infinite loop space BCokerJ, which is the basic building block (and the relevant space is different used in Em that from operad out to be the classifying 2) (away from and turns BTop for BSF and for I): is ring spectrum Em of the resulting the zeroth space chapter sp:-.erical fibrations for a suitable stable space for j-oriented X 2. equivalent to BO have two infinite loop space structures We thus BSF splits BCokerJ show that We also BImJ as X ring j. spectrum E, on BO corresponding to two machine-built connective spectra. If we when p, and that, at an odd prime localized as an infinite loop space are to take these structures seriously, then we must prove that the there is a (non-splittable) infinite loop fibration B Coker J. Im -tB -t 2, BSF J at obtained from the periodic machine-built spectra are equivalent to that contains infiruts loop space tneory. Chapter of pairings in M a theory spectrum by killing its homotopy groups in negative degrees. nott This is used to compare our machine-built spectra of algebraic K-theory con- boxes have not of black manufacturers (Other yet studied such to the spectra constructed and Wagoner by Gersten [30,79]. follows slstency the required model, For our first problems.) proof is a sequel to [45 and 461. Logically, this book However, I looping and rel~tion delooping. between a commutation from For our the de- Thus proofs. modulo it self-contained make have tried to the reouired sccond proof follows directly from the ring model, and finitions of operads spaces are recalled in 1, and the TIIS Em we rely on a characterization of the In both cases, structure. 5 in VII cited papers are stated without proof main results of the connective spectrum associated to a periodic space which only makes reading of may find reader Nevertheless,the a preliminary 1-3. ..ense of special features of our new construction of the because helpful, [45] leisurely contain a as they the first three sections of ,table homotopy category. models are necessary can be That both While behind explanation definitions. the basic of the motivation most %Seen in the orientation sequence clearly for k0-oriented stable a full understanding of the constructions used in VII requires pre- This is a fibration sequence of infinite loop spherical fibrations. liminary reading of and 8111, the pragmatic (and trusting) [45,59 *;paces of that chapter reader as existence statements the results may regard + BSF + BO + SF B(SF;~O) €3 the internal intricacies of which derived by means of a black box, is derived chapter from on SF coming structure E, vhrch by use of the can safely in the be ignored chapters. applicatiorsof the remaining by the second model. together the ring structure on kO E, given I with It is to be stressed, all of our applications howe~rer, that first model !'he is essential to relate this sequence to the natural is an infinite the mere assertion that a given space go beyond which +SF level. ]:SO nialp on the infinite loop Many of our applications space depend on special features loop structi -- the new multiplicative #)f chapters V deriva- around this sequence, and its and VIII center various consistency statements, flexibility in the choice of raw I ion really seems to require every bit of our general abstract mach-

7 1. J functors simpler consistency about Bott much the even That question inery. the is trivial altogether fact not is by indicated that periodicity prove result to well equally will apply the our proof second easier lifting cited above. about (additive) Brauer [19], Boardman and Vogt introduced the concept of $-functor. In Beyond intended to give a coherent its new results, this book is BFV certain show to purpose was Their collections of spaces, such as how descriptive most to opposed (as computa- important the of account or product spaces V produce E spaces and inner on indexed BTopV, w tional) infinite loop space theory. of applications study thorough A 03 limits over passage by spaces loop infinite therefore to C R . V section In 1, of in appears space Em t261, and spaces Em ring of homology the of part this of a detailed exposition give we terms in theory, reformulated their of results to applied are volume present the where the and theory of An informal the of summary we [45,§ a give particular, In 11. of defined in as operad actions and operads classes. of the characteristic study which is given in [49], and in one material both that volume this and their homogeneous spaces systematic the classical groups discusssion of intuitive gives an extended of area general this to introduction as as 9-functors -functors. d of morphisms and maps Bott display the topology. section 2, In the two-sided geometric we relate construction to bar is people many It the to debt my acknowledge to pleasure a very the that the effect to Vogt and Bpardman of rederive and -functors 4 theorems I Don to details owe Anderson, me this with who have helped work. the related classifying PL spaces and homogeneous spaces are Top, F, and Stewart Priddy, Liulevicius, Arunas Lashof, Dick Fiedorowicz, Zig spaces. infinite loop brief ad (We shall use the arguments based on hoc Larry Vic Nark Steinberger, am I Taylor. Dick and Swan, Snaith, to particularly indebted Ray, and Nigel Quinn, Frank coauthors my triangulation theorem to handle the of these sections ) of material Most PL. for their ideas and insights, to Ib Pladsen for J$rgen Tornehave 1972. preprint form since circulating in dates been has and 1971 from key discussions of 2-primary phenomena and correspondence about Vogt's Boardman own account of their theory has since appeared [20]. and various of this work, and to aspects Frank Adams a of proofs for are quite ours, from different and there details of choice language and Their number of lemmas and technical mucb other help. I owe a special overlap. very little is needed The in present be will language and results to debt of Jim careful his for Stasheff manuscript entire the reading book the rest of this . his many suggestions and +or its improvement. thanks Finally, my to Maija May for preparing the index. Linear and classical groups isometries 1. Let and the category 9 compactly generated denote nondegenerately of based weak Hausdorff spaces. let $, operad an For denote the &[y] of category c-spaces [45,§ 11 (or VI, $1). an operad To an action of obtain on the infinite classical groups and

8 an and the it, = pa that isomorphism such injection is 6 Then . e! = ) p(e related spaces, is conceptually and notationally simplest to pass from a it I 2i Define of the first summand. a path Hi:I + (RC0,Rm) identity the from a related operad action by an to $ x. a certain functor defined on category to :I -4 (V,V 63 normalizing rr and define a path H from i' to i" by V) are Vogt [19,20]. andq due to Boardman and The definition- a?Ld prapertiies of 3 2 the obvious linear paths $ the category Define linear as isometries of :i. Definition1 tf'! t (i-t)f! = (t)(f.) . and te t = Gl(t)(e.) (i-t)e. G are or finite dimensional countably infinite $ The objects of follows. JJ J 1 2i 2 finite dimen- real inner product spaces, topologized as the limits of their + H: I X by $ and define (V, (V, Rm) Rm) $ (v, Rm) dl E y Fix $(v, W) from V to W are linear sional subspaces. The morphisms (compactly given generated) is the + W, and (V, W) V isometries a that the direct sum Q: XJ + d) topology. Note is compact-open is commutative, associative, and unital (with unit continuous functor and Then and H(0, k) = k H(1, k) = p-'i"y, which is independent of k. (0)) isomorphism. to coherent natural up to and a $-functors now define We 9. from -functors functor Definition by 1. 2. Define the linear isometries operad X-spaces. m direct z(j) R of the copies j of sum = S((R~~,R~), where (Rm)j is w) Definition 4 -functor (T, An is 1.4. a continuous functor standard with its by specified are data requisite product; the inner T:J + and continuous commutative, associative, with a together w: transformation natural T T X T fB (of functors -+ -. X .! such .r) Qgk). ~(f:g~,...,g~)=fo([email protected] gi€ f~ x(ji). x(k) and (a) that 1 identity map. the is E q(1) (b) if and TV then the basepoint, is E T e 1 (0) (a) if x ~(j). s f for i(q) = (id(y) (c) E y and zj. . (R~)' i r TV 63 = {0)) T(V E x = 1) W(X, be is required to a sub-operad of the endomorphism other words, In and if m, < V' dim V", 63 = V' inclusion, the is V + V' i: if V (b) m Rm (where has basepoint zero). of operad R is (a), (which, by TV + Ti(x) = w(x, i), by given then TV' Ti: The x(j). on freely Z. following acts verify that It is trivial to J of basepoint the is 1 where a is a homeomorphism TV") onto E isometries lemma therefore implies that is an operad. Recall that m closed subset, and be isomorphisms. not need finite V' runs over the TV' lim as a space, where = TV (c) -. d. is contractible for Rm) all inner product spaces (V, V. Lemma 1.3. inner dimensional sub of V. spaces product iL e;l ortho- be {f!,fl!} and If.), i), (e.1 {ef, i), ir Let Proof. J JJ we write j, i l I 1 w the Whitney sum; for x. E TVi. call We m Define V respectively. Rm CB Rm, V, and , V Q normal bases R for w(xi ,..., x.) = x fB ... Q x.. Amorphism @:(T,w)-(T1,w') of J 1 J = a(e.) by = Rm + Rm a: el p(e2i-i) by Rm [email protected] + p:Rm and define eZi Q: 3 is a continuous natural transformation -functors T + T' which com-

9 construct To only construct 4 -functors. We we x-spaces, need [ -functors. dl of the category denotes ] 5 mutes with sums. the Whitney $ we need only study finite-dimensional -functors construct to that show next 1. category has ] [ 3 The (i) 5. finite products; if Remarks isometric isomorphisms. inner product spaces and their linear then natural transformation, is the interchange t: T'X T -+ T X T' Let be j n. n < m, the full subcategory of $whose Definition 1.8. (T X = . X t X 1)) T', (w X wl)(l X w) (T, (TI, w') con- of the graded subcategory be & n-dimensional, and are objects let category the Similarly, products. fibred has [y] mX 4 &): $ Note that the functors the the union sisting of of m+n bP -+ n' function space $-functor and d-functor (T,o), define the an 3 E C For (ii) w) + $ . An J*-functor (T, together define a graded functor &): Q,, x T)(V) F(C, = the composite sum TV), with Whitney F(C, by F(C, T) together with a commutative, associative, + 4* is T: functor a continuous rF(C, W)). T(V tB TW) X X C, F(C -. TW) F(C, F(C, F(A~ TV TV) X that such transformation and continuous natural w: T X T + To 8 is an w) (T, if and If U the universal covering space functor is (iii) is 1 E T (0) (a) TV E x if and if the basepoint, then from induced is Uo where -functor, $ then (UT, Uw) is an -functor, 1) W(X, E x = T(V €3 (0)) = TV . PT = F(1, T) by passage to quotient spaces (compare [26,It % 81). Po on &) given (b) if V = V' V", dim V < cu, then the map TV' + TV by [f] =L! by ] [ + letting Define @: a functor Definition 1.6. x a homeomorphism onto a closed subset. + w(x, i) is @(a) morphisms, on T'R~ + on objects and TR~ = @: @(T, 0) (TRW, w) = in ] defined are -functors $+ of Morphisms [x evident way, and d, the where TR~ 8.: x(j) X (TR~)' + is defined by category -functors. j.+ of denotes the J w . . TR E xi and x(j) E f xj), &) . . . &) (T~)(x~ = x.) , 0 . . xi, .(f, ] 3 [ an is TI [ $, -+ Q functor forgetful The Proposition 1.9. 3 J categories. isomorphism of w. continuous by the continuity of T and 8. Observe that 63 commutes is J We must verify that an 4% a unique admits w) (T, -functor constructions specified with the remarks, where various in the previous Proof. TV define = must If dim V and m, we can to an extension -functor. J constructions these on are defined 26, I. [TI by [45,1.5-1.7 4.81. and -functors and $ for both for E TV' write shall we 1.4(c); for 1 8 x' the image in TV of x' Definition by is customary, we shall often As write T Similarly, we can and must w). when dim V' < V (since B) V' = (v')~ . TR~ x-spaces derived the 63 y TW w: and + T(V TVX 61 W) 83 by x = (x' &) y') 61 1 if x = x' 1 define Then a: Let 1.7. Remark RW+ be Rm a linear isometric isomorphism. y with dimensional subspaces finite for TW' E y' ard TV' x' 1 €3 y' = E by operad the of xa automorphism an a determines Finally, W. of if and a linear isometry is W + f:V if W' V' of V and -functor, = is (T, If (xa)(f) an af(a-i)': (ROD)' + R~ for f E x(j). w) where dim TV' E x' with TV, define E we w, < V' x = x' &) 1 -+ the xa-equivariant an is TR~ homeomorphism, TR~ sense in Ta: then fl~':~' = = f' where , 1 &) (Tf')(xZ) f(V'). -+ (1) (Tf)(x) that 8 Tao 8 = (~a)'). .o ( fa X j~

10 . C TVXTVXTWXTW =TVXTW - TV TV X TV f I(v')~ is contained in This is forced since the image of f" = definition f f", and we therefore must have = f'61 hence f(vl)I, W) Q T(V x W) Q T(V B) T(V W) 1) (T~I)(x') = (~f")(i) Q (T~')(X') = 1, 8) fl1)(x1 Q T(~I = Q (T~)(x) equality holding since Tf" must preserve basepoints. It is straight- simply amounts to the validity of the the last formulas so constructed, is indeed a well-defined (T, w), that to forward verify (fgf-l)(fg'f-l) = and fgg6f-I hhl = (g 61 h)(gl ggf h') 61 B) -J*-functors extend uniquely to -functor. Similarly, morphisms of d composition a b\ (v, w), g, g' a TV, and h, h' a TW. On TR~, the f for of morphisms passage to limits. -functors .! by induced to (internal) Whitney sum homotopic from the will be product the 31. [ Q and Je [ ] categories identify shall we the Henceforward, ( see [45,8.7 or 46, 3.4 1). When, further, TV is action of a the underlying construct -functors but shall only of speak We shall tV + tV, the inverse homeomorphisms sub topological group of the group of The -functors. will be basic to the applications. following remarks TV will TV i: define a morphism of + map 9; by virtue of the -functors Remarks 10. For many of the I -functors (T, w) of interest, the points i. (fgf-i)-i = fg-If-' and tV * tV, for some space = h)-' Q (g h-'. Q g-i derived from) maps will be will be (or a TV x functorially tV the basepoint of TV will be the identity on V; depending the classical groups and their homogeneous spaces define We now dim V m, W) = dim W < (v, a point f a d of map tV. when Moreover, as systematically Let K denote $ of the normeddivision -functors. one and we have tV * tW will tf: determine will a homeomorphism numbers, R, H (real numbers, complex or or quaternions). For C, rings as regarded V [email protected] denote VK a real inner product space V, let a (left) TV. x for tfoxo(tf)-' = (~f)(x) a R J regarded 'VK denote VK let K, C J space for inner product and, over K (i) Henceforward, by (2) and notat- replace formulas generally shall we the ionally simpler expression as inner product over J. an space classical G, we undkrstand By a group topological of category the to j. from functors the following of any one a W). a f and TV d. x for (V, = fxf-' (~f)(x) thus suppress from the notation both the passage from f to tf and the We the It will often be restriction finite dimensional subspaces. to required (under composition) monoid of the monoid TV is a sub topological that case R R C S0(Vc) C 0(Vc) U(VC) SU(VC)C TV X TV TV C: * that the composition product It will follow -C tV maps of tV. n defines a morphism of J -functors. Indeed, the commutativity of the dia- grams a classical vertical inclusions are all of the form h he -c i @ g. ) If G is g

11 of n, of G (in the sense subgroups < i < i are classical and group if Gi, n is G/.X defined to be the Gi above), then the homogeneous the lattice space In each fact, by passage to orbit spaces from a map p is induced 1=i 2 2 which is given -+ objects by functor on 7 form of -+ ) p: the H(V ) RG(V n = G(V.)/ i=i x G~(v) . (~4% G~)(v) t 1. 2 S 0 and H(v~) g E , = p(g)(t) ga(t)g-la-l(t) n is induced passage Gi)(f) by (G/X m, < W dim = V dim E f For W), (V, a(t): Here V' + v2 is a linear isometric isomorphism of the general form 1-1 K K i n + g where E for G(vn), wn is vn ?: from orbit spaces to g -+ Pg(f- ) E v,vl for V a(t)(v,vT) = (val(t),v1a2(t)) K with induced sum Whitney With times. itself n by f of the direct sum of K. in the relevant one norm field elements are a.(t) where the ground from the composite passage to orbit spaces For example, w)~), B3 G((V > G(v) G([email protected] wn) G(wn) X G(vn) U/UXU = RSU + R(SU/exe) p:BU= n shuffle 4 an is u) Gi, -functor. isometry, each the evident is v where (G/.E is e-mit. so determined by a (t) = emit and a (t) = Explicit definitions of 1-1 2 1 and H. G., maps orbit spaces define of a morph- the evident C then C G H If be The in the real case may required a.(t) found in [21 and 251. the 11 n n ism of + the of inclusion the G, H= when ; Gi X G/ $ H. H/.X -functors of an 3 -functors calculation is from a morphism is p easy verification that each 1 r=i i=i n n not needed). u.(t) are the for explicit (and expressions maps the of form the fibre X G./H. in G/ X H. is also of -functors. a morphism the By i=i 1 1 i=i 1 that is The point W), (V. $ f E if a.(t). with commutes fz then G, we understand the morphism of universal bundle of a classical group use the natural to In iterate the Bott maps, it is necessary to order J? -functors n:EG= G/~XG -+ G/GXG = BG maps below 6: G +RBG for G = 0, and 2.4 2.3 Propositions Sp. and U, G in the = G2 and G, or e = framework obtained by setting n = 2, G i become maps composites of show these 7-spaces, to passage after that, that the Remarks on 1. 10 show product and inverse map each above. are hornotopy equival- X-maps of and homotopy inverses of x-maps which classical -functors. are morphisms of group G ences. of our 1153 stable category the arguments definition our As in and Spin by letting Spin(V) be the universal cover of SO(V). We can define discussion. following complicate the not will inverses these clear, make will SpinC, Alternatively, be can pinC and Pin, and also and Spin preferably, 1.6 Definition yields a natural structure of on the homo- q-space co n as -functors by means of their standard descriptions explicitly described geneous spaces G/ on of these Gi)(R X (G/. = Spin, Gi ), etc. Each 1 I= 1 i= [12]. Theproduct of the Clifford algebras of inner product spaces in terms grouplike is spaces a group) and is thus (no infinite loop space by is an and inverse maps on these groups and the usual maps between these groups (or 3. 2 below). VII, 2.31 r46, Certain of these spaces are also infinite loop -functors. of classical and are then groups morphisms spaces by Bott periodicity. these structures, To show the consistency of of the Bott maps may be regarded as a We observe next that each spaces 2 = X be one of the d, d entering into Bott periodicity of period let 2 8 3 -functors of the form of morphism or d = 8, such as R BU r BU X Z or $2 BO -- BOX Z. The x-space

12 18 A A(yg, x) = A(y, gx) such that the intended choice of p and [45,9.2]; on structure X (or infinite determines a connective coordinatized spectrum should clear from the context. be G-maps the By and 11.101, [45,9.8,9.9, that X. to equivalent naturally is B X such {B.x) = B X sequence) loop 0 m and E:B(G,G,X)+X E. :B(Y,G,G)+Y The iterated Bott P: induces X a weak 52% homotopy equivalence + map We ;Ire retractions with right inverses the evident maps T. deformation spectra. As will of be explained in VII. B~% 3.4, [46,3.1] - BmX B,P: Lkgree p are identity maps, and to abbreviate B(IY, y, p) = BIY if y and d of B X the stable category, the 52 + 52% B in a map gives zeroth level m m the similarly in other variables. a%. identity there- we composition, By map which of is equivalent to the is is G. of classifying space the standard If *) G, B(*, = G BG d fore have a map B X is - 52 BmX, the zeroth level of equivalent to which m a weak homotopy + S2B equivalence. &:.G is G group-like, inclusion the natural be intuitively obvious, and will is $3, +a%. p:X II in proven rigorously It If, aspherical, E' with G-fibration a principal quasi is B' - E' p': further, that the spectrum associated to the zeroth periodic connective spectrum with Lhen the maps determined space X by in the stable P is characterized, up to isomorphism B' xc) B(E1, G, b BG <- conditions. precisely these by category, display homotopy equivalence between a weak 7.71. and [47,8.7 BG and B' j: H -F G of grouplike topological monoids, define For a morphism left and right the spaces" by "homogeneous 2. PL. bar construction; The F, Top, and G). = B(*, H, H\G B(G, *) and G/H = H, bar geometric two-sided The role in a central play will construction 8: the FBj FBj, -+ G/H denotes equivalence homotopy a weak is There where be used notations will following throughout theory, and the the later our theoretic fibre, sequence a quasifibration and I~ornotopy chapters. Let G be a topological monoid the a is which of element identity X let and nondegenerate basepoint Then right G-spaces. and left Y be and BHL BG G/H& GG HL G,X), B*(Y, space there topological a simplicia1 is the of space which n-th the H\G. for hold statements same By symmetry, 81. 8. [47, Y is will be .l] 11 [45, Its geometric realization $101. [45, X G~ X denoted X [46,$3], B(Y, G, X) is a explained in As Now any be 6 let operad. write always shall We X). G, B(Y, G c-space G, and X are -spaces and the product Y, and unit of when +B(*,G,X) and B(Y,G,*) + p:B(Y,G,X) q:B(Y,G,X) is Such a G ("-spaces. of are X and on Y on action its .~nd morphisms - and * + X G-maps trivial the from induced maps the for * where *, Y the homotopic to is product monoid its c[T]; in sS,id beamonoid to product the is G-space. p and q are quasi-fibrations with fibres X and Y one point l443.41. the of any con- maps given all When in the action of by given when is G are and a topological group is when G G-bundles grouplike then c-spaces, of morphisms are paragraphs the previous of t;Lructions shall write We and 21. 8. 6 [47,7. all Y are as interpreted derived maps are -+ I X X (where homotopies HO + E(A):B(Y,G,X)+Z E= and T=T(~):Z+B(Y,G,X) via 6-space a is &-map -F !naps a of fibre the where and ,Y) F(I X by a for map the maps induced Y a map X and by + Z p: X + :Y A Z X X

13 this [45,1 most of the maps above, For statement follows from the fact .a]). point only The well-defined. is G,X) B(Y, easily verified that is It &-spaces simplicial from a functor defines that geometric realization continuous B(Y, G,X) is because that the functor is worth mentioning set if continuous is spaces the of maps of simplicial geometric realization calculations the remaining maps, easy direct to (?'-spaces [45,12.2]. For the subspace of product over n of the a C from to D is topologized as as in is handled, are -c G : required. where [46,3.6], RBG 5 from spaces of maps first x-space Dn. to Cn the obtained by Clearly We next that precisely similar statements apply to show -functors. to 8 applying and then G,X) (Y, the B of functor the applying functor in A monoid-valued $) -functor, or monoid Definition 2.1. [ 71, 8 applying first to 7-space obtained by coincides with 1.6 Definition the is -functor identity the monoid, topological a is GV G $ an such that each G, and Y, B. X then and applying of GV element its is the a define GV + GV X GV products basepoint, and of In first the of material the few all paragraphs 5, view of Remarks 1. a is GV each if group-valued be to said is G -functors. 3 morphism of (where of can this homotopy a section -functors Q of terms in be rephrased GV + GV maps the inverse and topological group of define a morphism a morphism J-functors means -+ T1 of 3 of -functors T between maps G is said to be grouplike if $ left A a group. is GV -rr -functors. each 0 f T of a map of the fibre for spaces, 4-functors where, just and ,TI) F(I - as on of a monoid-valued J -functor G an action a morphism -functor X is again is It straightforward to verify product). fibred evident is the as defined of is +XV that the X XV GV map G X X an Q -functors + X such action of that all constructed maps are morphisms given maps -functors when 9 all of V. GV on XV for each in the following statements. results summarize We so. are acts d -functor which be a monoid-valued Definition 2.2. G Let + H If grouplike monoid- of a morphism is G j: Proposition 2.3. from left and right on the an d-functor $ and -functors Y. X Define is of a quasi-fibration sequence valued Q -functors, then the following E V For follows. as G, X) B(Y, f: V for and in $ + , Vt a morphism -functors $I define BG. LG/H&BH G HL B(Yf,Gf,Xf). = B(Y,G,X)(f) and B(YV,GV,XV) B(Y,G,X)(V)= 8: of + FBj -functors. and RBG + G 5: weak equivalences are G/H the G,X) is defined by B(Y, composite maps on Whitney The sum 1.10. $ ~kmarks -functors by classical define group-valued groups The Proposition When G is a classical group 2.4. L. -functor, maps the GW, XV X XW) B(YV X YW, GV X BG G/GXG --%-+ B(G/eXG,G,*) a w) B(w, (0, I -functors $I determine between the x-spaces of equivalence a weak of W), W)) X([email protected] G([email protected] W), B(Y([email protected] natural two classifying of spaces G. and associative is the commultative 5 given natural homeomorphism where by 11. :45.10.1 and 51.

14 Henceforward, we shall work relative to a fixed continuous We pass from -functors to 1.6 q-spaces via Definition and from tW W). By B the $, -+ 7 such that t(V = tVh t: sphere-valued functor to infinite loop Top, F,SF, X-spaces spaces via [45and46](or VII§3).Thus of product, continuity, commutativity, and the smash associativity is itself t STop, the classical groups, Spin, of spinc, related groups and all and their an $*-functor with the natural projection W) as Whitney X -(. t W t(V (B tV spaces spaces classifying are infinite loop spaces, and all and homogeneous sum. We have the following list of monoid-valued d*-functors which act of natural the maps between these spaces maps. loop infinite are from the left on t. perhaps It remains to consider PL and related spaces. One could (Y a complete geometric develop the same with categories introducing by theory Let F specified $*-functor the denote by 2.5. ExarnpIes # objects as f[ k- but with simplicia1 sets of morphisms from V to W, the tV) f: V for tf) ~(tf-l, = FV , F(tV, -. = and $f V' rU would piecewise appropriate be linear which of simplices and piecewise X the Whitney sum ~:Fv FW -. ?(v f3 W) given by the smash product with w differential fibrewise homeomorphisms X A would PD here W; X Ak -. V of sub by F of STOP and restricting maps. Define TOP, Q*-functors F, SF, k required to PL relate to order in be through to 0. I have not attempted go homotopy attention to based equivalences, degree one equivalences, homotopy the details. Define tV. be to e homeomorphisms, degree one homeomorphisms and of w of I?, eV the trivial (1). When tV is the one-point com- Q*-functor sub = to consider be would approach second such llPL" -functor 4 the A and SO pactification sub -!)*-functors of Top and V* of V, 0 become the is "PLV" piecewise the based of consisting FV of subspace that of one-point compactification maps STop via the twisted . adjoint Moreover, of homeomorphisms linear wrong homo- has "PL" Unfortunately, v*. the from Pin, representation of [12, p. 71 defines morphisms of $*-functors communication), topy type; according to Rourke (private the inclusion of SU and U , 0. V: = tV to r, pinc, Spinc, and rC When Spin, become "PL" in conceivable least Top at is It equivalence. a weak homotopy is $ *-functors of STop via one-point compactification, and similarly sub on lhat homotopy a larger the right have topology "PL" exists which does Sp when for V: = tV j,-functors these of all [ld, and 1.10, Remarks By . type. Z monoid-valued and all but SF, F, are group-valued. and F are these difficulties, PL that to recall was ignore to is approach Our to and largely introduced in study the triangulation problem, to observe order pass We from Proposition to 3 -functors by j*-functors 1.9. w completely PL/o, etc. are determined of types homotopy the that BPL, PL, F, ""FR~ = 'ij(n), ?R~ = Write and similarly for other J -functors. spaces by the solution to this problem and by the infinite loop .ts loop infinite G(n) to be the space of homotopy equivalences of all define to usual is It Sn-l on already derived structures space Top, ETop, Top/O, etc. the equivalences. based homotopy subspace of and to regard F(n-1) as detail, In 3), = is a 2-connected Top/pL K(Z2, note that we Top/0 G is the homotopy = G lim G(n) is then equivalent to F. It is clear that ~op/O (TO~/O) IT hpace, the natural map and + TO~/PL induces Z2, an = some shall we but more natural space to consider in geometric situations, 3 Recall spectrum (n-1)-connected one only is that there isomorphism on rr3. work with it is our F since this space which occurs naturally in theory.

15 th a n) (up to isomorphism in any good zero- space with n) X(Z2, K(Z2, Coordinate-free 11. spectra non-trivial only there is and stable category) one map (n-1)-connected from an into TO~/PL, it is con- T n). 'X(Z2. Thus Z2 = E however with spectrum n loop space, and the structed geometrically, is just K(Z2, 3) as an infinite defined T. A spectrum T is usually spaces to be a sequence of unique non-trivial map an infinite loop map in precisely TO~/O + TO~/PL is Titi. Rm be } {e Let the standard basis and for and maps + Ti Z u.: n one way. the think of the sub- compactification tRe of the one-point i-sphere as n of Re space notation allows us to describe T Then a change of Rm. Now following spaces (or rather homotopy types, since that the define n T(R~) +T(R~"). tRe andmapsw.:~(R~)h asasequenceofspaces is In that any permissible it case, each clear determine). is all our data 1 i+ notion Thus the usual to of spectrum implicitly basis a fixed chosen refers homo- named specified the the geometric construction of yield must space RCO. for Many in spectra, of theory homotopy the in difficulties very real type. t0PY particular associated with the construction of well-behaved the problems the fibre of the unique 3). map TO~/O K(Z2, + PL/O is non-trivial (a) products, smash Such suspension coordinates. of permutations from arise Top the composite of fibre the is PL K(Z2, and + TO~/O -c 3) SPL (b) permutations be thought of as can from changes of basis for resulting ROD, is STop. fibre of the the of this composite to restriction be and we shall see in [48] that the coordinate-free definition of spectra to is of BPL the fibre the composite BTop B~(TO~/O) + K(Z2, 3) and + (c) leads given here the of the properties of development relatively simple to a the this composite to of restriction BSPL of fibre the is BSTop. stable category. homotopy to the ) . 5 3. VII. by Top (Here BTop is equivalent B delooping 1 with our present concern is However, more than just stable homotopy (d) + BE. -c BTop BPL composite the of fibre the is E/PL In theory. define to order essential is work spectra, ring E a in to it infinite loop map is an infinite loop space, and the of an fibre Clearly m spectra (omega) of category before passage even properties good which enjoys it follows that each space we have constructed has a well-defined infinite point The is that these spectra have very rich internal to homotopy. space loop spaces these between natural maps the of all that such structure is homotopy to passage upon lost the which of much structure, category. related PL/G and constructions maps. are infinite loop Similar handle stable constructions previous best homotopy the of used in The spectra for groups maps other classical G. derived are (or are categories those T from) spectra CW-spectra, namely such that inclusion. a cellular is u. each a CW-complex and is Ti loop Obviously such rigid structures cannot possibly be related to infinite e as be will spectra our homotopy, and to before passage spaces cell-free f unde- as coordinate-free. Restriction to well is in any case CW-spectra

16 It will be important to keep in mind the old-fashioned distinction be- sirable nature are not closed under in occur since CW-spectra seldom and t the 8 former and for internal direct sums; external tween we write and product and loop formation as constructions useful and such simple of sub inner two indicate that to W 1 V write We product latter. the for In the loop by given be will category, desuspension stable our spectra. will the ; notation t are space product inner a given of spaces orthogonal I functor. spectrum between orthogonal subspaces and will thus carry orthogonality only be used spectra, Wedefine coordbate-free prespectra and back show how to pass 1 an implied hypothesis. as spaces, relate and spectra, to coordinate-free and and forth between prespectra, 1 subcategory of d* the objects of which J Let *(Rm) denote the full section coordinatized i. We define the stable homotopy category spectra in l spaces the finite-dimensional sub inner product are of Rm and let and spectra, com- localizations and and spectra, connective ring discuss I Let ha homeomorphisms. based based spaces and of category denote the proofs pletions in these sections; of spectra in section 2. We shall omit most I Since to associated category homotopy the denote a. category a topological in [48] and are largely irrelevant to our the missing details may found be two has precisely W) (hJn)(v, O(n), homeomorphic to is W) $&(v, elements book we consider cohomology theories and 3, section In . this in work later simplification 10 Remarks the of 1. below a possible indicate 0. n > if between relationships precise the of analysis pedantic a rather give periodic definition of the following connective "periodic and spectra, periodic spaces, spectra". Although exploited-in a wholly different way, the of using linear idea m u) A (T, prespectrum teg + ) $*(R T: a function is 1.1. Definition due isometrics study the stable category is to to Boardman [18]. Puppe [56] induces a functor T: hj.*(~~) -+ h$ (on rnorphisms) which objects and independently came to the idea of coordinate-free prespectra. I satisfy T(V t W) for which based maps u: Z~TV -z VJ,. W together with the following conditions. spectra and prespectra, Spaces, i. is W) with SlWT(v + TV z: map adjoint Each inclusion an + (i) in as and, 1.8 I. from * a of definition Recall the a con- fix 2, § I image. closed and = (0) that t SO 3 -+ * d t: functor sphere-valued tinuous such : V L Z 1 W Vl where , d The following diagrams commute in (ii) the practice, compactification In tV will be one-point = tW ~VA t(V8W). and = z~+~ Z'I;~TV TV ZOTV = TV of TV for some functor -r that 1.9 Remarks in see shall we and , $ -+ :\S restriction to identity functor the T would in no real loss of result based Y) denotes the space of F(X, maps X + Y generality. Recall that V to be and define the suspension and loop functors based on The following diagrams commute in h$ , *(v, where f J VV), (iii) and W VI €9 w'), ,(w, : W1 VIJ. g F(tV, zVx = X tV and SIVx = X).

17 The use of the homo- specifications way. merely codifies these in a coherent m ) a systematic topy category h'$*(R letting for device can as of thought be signs, linear isometries keep track of changes of coordinates, and such like complications in the usual theory of spectra. fW) is a homeo- (T, is said to be a spectrnm if each G: u) TV + Q~T(V to define tempted One might be (TI, -+ a) of of maps consists prespectra ul) 8: (T, requiring T to be a functor prespectra by morphism A morphism. be and u to *(ROD) but 8: not would definition such a TV categories, homotopy of use without natural, such T + T1 is a natural transformation + that T'V 8: of functors in 9 diagrams following the and such that commute for spectra coordinate-free of allow-the construction of or from prespectra -F hd h4*(RW) spectra. spectra from coordinatized VL w: 3 , next make We categorical interrelationships among the precise for @, and ,d (as was done in [43] ordinary spectra). we Observe that @ + defined d) and + d have zeroth space, functors or forgetful, full sub- @ denote the category of prespectra and let 4 denote its Let by (T, = To on objects ~(0). u) + the inclusion functor. denote v d : 8) -+ Let category of spectra. Definition 1.2. Define the suspension prespectrum functor be to is @' and primary interest, is d of as regarded The category 1 -F by letting 3 Z: spectrum), The pair of terms (prespectrum, category. a convenient auxiliary 3 (zWx)(v) = zVx and (zmx)(f) = z% tf I/\ is distinctly preferable to the older pair by Kan, which was introduced letting by and 9*(RW) on of f morphisms and objects V (spectrum,Q-spectrum) since spectra study of fundamental objects the are of homeo- The use naturally give rise to spectra. and prespectra since Ir = 1 : zWzvx + xvfw x for VLW. of in spectra the definition equivalences, homotopy than morphisms, rather --Zvxl. let FO~~$:X+X', 1 have We theory tothe applications. the and convenient in both is essential 1.3. Lemma generated the by free is prespectrum zC0x the SZ-spectrum. classical for use little an the notion of is a unique map @ and 6:X E To, there + for is, that X; space T Granted theory of spectra, it of desirability a coordinate-free the ," th of zWX fi -. : prespectra with zero T map 6. sensible clearly is of the finite-dimensional subspaces of think to as R~ the associated spectrum functor Define c? + am: 1.4. Definition at a prespectrum least, to consist of spaces Thus ought, an set. indexing k . @ E u) (T, Let follows. as $ have identifications We insist and reasonable obviously is it to W), f T(V + Z~TV u: maps and TV TV TW homeomorphic to that be if = dim W. Our definition dim V

18 31 30 A E and . e e X 3 (X, EO) g d (Q~x, E), 3 Wl L V and we define VL Z for and relate and spectra prespectra Finally, we define coordinatized the coordinate-free variety. them to taken with respect limit is to the the where inclusions 1.7. Definition (0) = A0 C A1 C . . . C Ai C . . . , be = {A.), Let A m of subspaces of increasing an sequence R fim with , = Ai. Let Bi i, 0 required homeomorphisms For VLW, the orthogonal the of A. in Aifl. complement denote {Ti, ui} prespectrum A I A indexed by and maps T. spaces based of a sequence is + W) (nmT)(v) -+ g: fl(nmT)(v b. inclusions are ITiSl S2 + T. g.: the that such TiSl tBi+ adjoints 11 To are to passage the identifications limits over obtained by from Z (*). images. with closed is each a homeomorphism. u. if a spectrum is { ui] T~, (Q~T)(v) (nmT)(f): (CZWT)(~') for f: V + V1, choose Z which -c define prespectra A ui} -). {Ti, IT;} of {ei}: is a sequence map of based maps {Ti, W and W' be the orthogonal complements contains both V and V', let bi - H.: T. T1 and @A Let 8.. oZ such that = u. Oitlo u! denote the i 11 11 and, to up one is in 2, and observe that there homotopy, V1 and V of of of spectra. A and ,.ategory full subcategory by indexed prespectra its f Sg such (through homotopic is that isometry one only g: W linear + W' by letting $(T, u) = {Ti, u.}, $: Define a forgetful functor [email protected]+ @A where m T)(f) to the identity of 2. The isometries) homeomorphism (n required bi = 'I'. u. and TA. = u:Z TAi + TA i+l ' maps the from Z'LZ obtained is passage limits over to by With modifications, the previous definitions and lemmas trivial Z1) T(Z awsSz' + + ZI) + Q~~~~~(~ spectra. and to coordinatized .rpply prespectra The following would result T1 given by composition with t(g-l+ 1). For a map 8: T + of prespectra, prespectrum level. I,c the false on (nmT)(v) passage to nm8: -+ (nmT')(v) by limits from the maps define Theorem The forgetful functor 1.8. ,d is equivalence an $: &A + + awe: W) -. nWT1(v + W). CT(V ,,f categories; that is there a functor is, & + AA there and are : $ 0 11.ttura.1 isomorphisms T(V TV inclusions W) lipCZWT(~+ {0)) + The Lemma 1. 5. = define a map of is vnmT + free : L the spectrum nmT and prespectra, T T -. vE, there T; that is, for E 6 d. and 8: generated by the prespectrum Y 1.9. I(r.rnark u that nontrivial some for m additive TV = tV Suppose - m 8. = 01. such that spectra of E -r T 8:n map a unique is tttnctor TR~ countably infinite has . $ dimension and we T:$ -+ Then d + (Q m X) and define QX 0' = nmZm: = Qm Definition 1.6. Define isometric g: isomorphism TR~ 8tt.ry choose an + deter- Visibly g the free QmX nnZnx. li_m is to is DX Observe that then homeomorphic category tjtrnes isomorphism between the an d~ defined by use of t and and X, by generated spectrum adjunction an have thus we !It$. category ,dgA defined by use inner the one-point compactification of of

19 letting to equivalence, the category Therefore, up d is independent product spaces. Tf) TV) F(X, T)(f) = F(1, and (X, F = T)(V) F(X, (of of the specified form). of t the choice J*(Rm) and by letting the structural of on f morphisms V and objects that use of linear isometries in explicit implies 1.8 Theorem 1.10. Remark the of adjoints the be u composites maps a spectrum quite our definition is Definition in unnecessary, and the details of ' isometries conceptually I find the introduction of is this why show 1.4 the case. F(X,TV)[email protected],Q~T(V+W)) ;I~F(X,T(V+W)). 3 IV. prespectrum level on particularly helpful, the 1.3), but the (compare then T). F(X, is so If T is a spectrum, reader throughout. ignore them to free is Lemma 2.4. For X €3 and vE). F(X, = E) VF(X, , d E E For = E 7 and T €6) , F(X, T)~ x F(X, T~). 2. The stable homotopy category Lemma 2.5. X For isomorphisms , there are natural (adjunction) 3 E "small" We require small smash products and function spectra, E @(TAX, TI) @ @(T,F(x, TI)), T,TI G spectra and meaning between spaces and spectra rather than between E')) and . A E' E, , A F(X, A(E, % E') (EAX, spectra. Z the Vth map -has E1 -c X EA 6: of El) F(X, -c E 8: adjoint Explicitly, define and T E f? , TAX E 55 by For X E 3 2.1. Definition EV EIV) fJ the composite the of -c in adjoint F(X, letting X)(V) = A] Tf = (ThX)(f) and TVAX (TA morphisms and V objects structural letting the andhy Q*(R~) of on f Let and K a space of a disjoint basepoint. K+ union the denote composites the be u maps , A both 3 Z and Y For 2.6. @ or or in define a Definition f homotopy h: -- fl YAI' Z h: a map - Z + Y f.: maps between to be 0 + Y hl that such the relevant category) h (in Note could that {i) A = fi. E nrn(v~Ax). by & = For E E$ , define EAX EAX + F(If, 2). well be equally considered Y a map as denote the Z) n(Y, Let and v nrn similarly to transport to that Observe can be used set of homotopy classes of maps Y Z. -c The preserve not any functor spectra prespectra which on spectra. does machinery The basic of homotopy theory, such as the elementary m spaces. products preserve smash and nW with functors Z 1%' Milnor exact sequences, dual Barratt-Puppe sequences and dual E Y , and 3 zm(yA X) is isomorphic to Lemma For X 2.2. E in well applies equally 2.2 Lemmas 1. I [48, & in and 3 imply 2.4 and , T l):nm(TAx) A L nrn( -- [email protected] (Q~T)AX and is an E X For (Z~Y)AX. 3 that d and d @-c functors zm, 4 am, am, v , and the zeroth space isomorphism. all Q are still free and are am, homotopy preserving. Clearly zm, m and T E B, by E @ , define F(X, T) 3. E X For functors after passage to homotopy categories. a we have In particular, Definition 2.3.

20 functor the stabilization to spaces from spectra. Qm as regarded be should natural isomorphism I Let of type homotopy the (based) of 3 in spaces of category the denote ?/ and XE~ n(X,E)=.lr(QmX,E), EE~. 0 I For X and Y ?I €3 , hd categories the Y); n(X, = [X,Y] E CW-complexes. 1 E For F(S = RE S' h E and ZE define , ,f E E). , = 2.7. Definitions are equivalent [48,III]. Analogous H& Ha for valid and are statements E by define the groups and So Q homotopy of = Abbreviate S m [48, XI]. r-,E n(Zrs, E) E n = = n(RrS, E) for r ) 0. and H& admits smash unital and a coherently associative, commutative, map A 0. 2 r for E n = E n 2.2, Lemma by sr, Q 8:E-cE1 Since Zrs be to ring spectrum a (commutative) Define 1. a [48,XI S with unit product r w rO to be .d a weak equivalence if n 8 is is an isomorphism for all in said spe~trum E together with an associative (and commutative) product r. integers a two-sided EA E + E with fi unit S + E. e: following lemma The play in role our study of will a vital and Brauer lifting in Bott periodicity gives natural maps R and Z adjunction between The $2. VIE . E + E:ZRE and +RZE q:E a ring of $ The product 2.9. Lemma map E induces a spectrum is theorem [55]. desuspension the Puppe of The following result a version E E. to Eo from A $) (again denoted q 2.8. all Theorem spectra E, For equivalences. weak are E and is coherently naturally isomorphic to 6 X Proof. For spaces X. E such L that and a functor Hd 4. h& L: There is a category H.L a relationship such A E indeed, 1; XI [48, X large smash Q and small between m carries L objects, on function identity the is weak equivalences to isomorph- a standard category. products Definition Via is stable good any of property XI]. [48,II and isms, property latter the to respect with universal is L and a map 1.6, the identity spectra. of E -c Q,EO +: map determines Eo of Its morphisrns are composites stable H .i the category. homotopy We call is The the composite required map morphisms in h of . & weak of formal inverses equivalences in h d and and Z the functors , HJ In categories, of equivalences inverse become R n r n E = 0 for 5 and if A be n-connected to said is E spectrum be It will shown in [48] we therefore write and r. integers for all x-~ = Rr to it if connective be is loop space theory is con- Infinite (-1)-connected. properties one could hope for and, despite its all has the good that H& obsenrations. we the quire re following and spectra, connective with cerned equivalent to fact in is different definition, of the stable categories wholly E -. is Lemma 2.10. If C and D are (q-1)-connected and O:D Boardman [18] and Adams [7]. is an isomorphism for n.8 that such HJ in map a q, i then all denote invert- formally by hJ from obtained category Let H5 the isomorphism. an is E] [C, [C, *: D] 8 morphisms of set the denote Ha in Z] let weak the ing [Y, equivalences and cofiberings 8. Up of cofibre to sign, the denote F Proof. Let - we Again, HA between spaces have or Z. and Y spectra or EE and & . [X,EO] [Q,X,E] , XE~ r

21 0 3.1), Definition (see El Hn [E, = r0E) Hom(noE, noE) = (E; H Proof. and skeleta, its a CW-spectrum, induction C by applying over placing 0 well-defined. d hence a ring map because d is is we = 0 that easily find niF] exact sequence, lE1 [C, the using HT E, [Eh = E) n noE, @ Hom(noE con- explicitly can be d that Note El. The . sequence Barratt-Puppe the by follows conclusion i 0. 2 for 0 0 to of application by structed the discretization the functor nmT a spectrum to one and, exists there E, For up 2.11. Lemma = Eo + noE. aOEO map and D spectrum connective 8:D equivalence, only one ~d- in E + Let Localizations and completions will needed in our work. often be E is il a ring spectrum, for 0. If such that n.8 is an isomorphism 1 T be to said is A Abelian group an that Recall primes. of T-local a set be 8 structure D admits a map is then that such spectrum ring of a unique if it is a module over the localization at of to and ZT Z T be T-complete spectra. of ring if (connecting) natural and the 0 = HO~(Z[T-~]/Z, A) homomorphism While the existence and uniqueness could be proven by Proof. - 00 (where z[T-'] is A EX~(Z[T-']/Z, A) is the localization + an isomorphism ;:R stable we simply note that the map techniques, TEO + E in ,d con- away Z of T-local be said is Y space simple to (connected) A T). from properties required structed in given that, and 8 of VII. has the 3.2 localization or T-complete if each n.Y is T-local or T-complete. A 2 yields the naturality of + E 8:D B1:D' -+ E' as specified, the and h:X T at space X of -+gT y:X a map completion or +XT is a simple HA? targets with arrows all which in , following commutative diagram in a T-local space such T-complete or that into isomorphisms: other than E are [xT, [X. Y] -+ Y] [GT. $: or Y] [X. + Y] A*: equivalently, or, Y spaces T-complete T-local all for an isomorphism is or Z that (with such = z/~z) P A oreach A,:H*(X;ZT)+H*(XT,Z T, pe Y*:H*(X;Zp)+H*(XT;Zp), ) T D follows A D assertion is connective, the about ring structures Since and unique), and equivalent is exist X ;iT (and y are isomorphism. an is previous the from directly lemma. where X and 2 denote of to the completion to and T at XT 2 X connective our that proof not only spectrum gives an associated Note P pa^ P' P and localization the p. X of at completion Localizations and completions & already 5 on gives . , it a functor such functor on H commute but not localizations and loops and products, fibrations, with (commutative) For a ring R, the Eilenberg-MacLane spectrum cofibrations, suspensions, completions smash and commute with wedges, a (commutative) ring spectrum. ?C(R, = 0) is HR A A A X + ~:XAX' y~ of T at completion the products. However, Xk is T 2.12. Lemma the then spectrum, a connective ring is E If unique equivalence. an a map E n -+ Hn E in H is which realizes the identity map of d E d: map 0 0 The completions just described are those due to Bousfield and Kan spectra. ring of not adequate [23]; completions of Sullivan [73] are the for our applications i i i i..

22 38 will topological treatment of localizations and completions A new VIII. in E be 0-connected spectra such that Theorem 2.13. Let D and it will are there not that [48], and appreciably completions be seen in given be be Eo + f: D let and ZT over type finite of are w*E and T*D (in a map the category Incidentally, Hf to analyze than localizations. difficult more -+ E : 6 a map of zeroth map the is p at localization whose HT) D (in P PP study to which in one appropriate equivalent is ) h?j localisa- its not (and the E 0 and either Assume that D 0 no have T-torsion T. HA) for each prime p E tions and completions since Postnikov towers with infinitely many non-zero is a finite D; where each ~el(D;)~ r Do and an H-map or that is f 1 never have the homotopy type of CW-complexes. homotopy groups the f of a map is Then zeroth map 0. E~) (2~2, lim - CW-complex and = of completions localizations and need also connective We spectra. cl CW- is Dq each where T Dq Tel a finite D Moreover, E. + D 6: if 1 literature, a situation that will treated in be nowhere the This subject is and exactly 6 map such one is there then 0, = E) (ZDq, c spectrum lim which spaces with verbatim In H1( , applies summary above the [48]. rectified in map localizes to the given 6 at each prime p e T. P replaced by connective spectra, the only twist being that the commutation of Let D and and w*D such that spectra be 0-connected E Theorem 2.14. a map (in HT) whose Eo be let f: and Do their implies commutation loops completions with products, fibrations, and are Z over type finite of T*E (PI with not with finite smash cofibrations, and suspensions, but still wedges, whose completion at p is the zeroth map of a rationalization is an H-map and c. A AA ,. Again, the completions at T of E y A y: E A + ETh and, E;, products. p-torsion (in Ep + bp:D map that Do Hb.). Assume no have E and either P 0 A. A where each in are isomorphisms . D$ is Hd CW- a finite AX of space y r, y: E X, + a simple for X fi E Tel(DOq) -- Do that or H-map an is f and T T P 1 the completion spectrum a ring is at spectrum a ring of T In particular, E a map [email protected] E. + D complex and 6: (ZD:, E~) = 0. Then f is the zeroth map of with unit and --% E PT and product 5 Dq is a finite CW- spectrum where each D =Tel D Moreover, if P 1 Ijm (ZDq, E) = 0, then there is exactly one such map ,d which completes at p LO given map 3 . the P The zeroth space functor commutes completions with sense that the the in of results complete the is irrelevance these of feature pleasant One A ET space zeroth of is equivalent to the product of the completion at T of terms the dis- be will As 0. to associated En let spaces > n for Dnand 1 terms [9 group stable the and Eo Anderson of results [48], in cussed lLm that the show 111 and discrete the component of the basepoint of book. this to relevant cases the in vanish = Z, integers EX~(Z[T-~]/Z, X~E~). When s E T-adic the latter group is the 0 0 Taken [48]. pair need the results from following Finally, we shall of and spaces spectra periodic Cohomology: 3. be- map technical hypotheses, a minimal under that, assert they together, E-cohomology define E, and Y spectra For Definition 3.1. the spaces map loop infinite completes to an which at loop infinite T-local tween by Y El [any, = E~Y z~E]. [Y, = loop E T is it'self an infinite map. p each prime

23 I I I I I\ 1IItIl 41 For is usually called the X, a space define = E~Q X; this is what E% WH~ Let and weak homotopy classes category denote the spectra of w X. of E-cohomology reduced terms be described by can , E% Hg of In of maps in Hh . I allow Inductive mapping cylinder arguments [43, Theorem 41 one to E-% if dim tV = n 2 0 and 2 = [x, if n EV] [X, = E% 0. replace prespectrum a weak T a weakly equivalent (coordinatized) pre- by of our treatment will essential be E% of This description to weak homotopy. spectrum, functorially up to use the then can One functor Rm chapter. orientation theory in the next analysis complete of homology A of the functor coordinatized 1.8 Theorem the (on 1.4 Definition + level) and of will be pre- the within theories HB by given framework and cohomology an spectrum in actual obtain to one a and A. Alternatively, can use Q w in sented of the [48]. machinery is available. Suffice it to familiar that all say level direct telescope construction on the to weak from pass spectrum pre- result, which is proven We shall shortly in need the following [48]. to spectra [48,XII]. spectra Either way, one obtains the following result. standard YR' and tV with coordinatization Yi = Observe that, oo, u V = the a spectrum E, coordinatized underlying the course regard of may we For of a homomorphism the restriction specifies 0 e [Y,E] to maps Bi E [Yi, Ei] a weak as V$E R-prespectrum. prespectrum EOY 1s E?ii . Theorem 3.3. category the from a functor L is There weak of pre- 0 i 3.2. Proposition For the map E Y E, and Y all spectra lim E Y + ci category maps to the homotopy classes spectra and WH~ of weak there and 1 i-1 i. E epimorphism is with Y to isomorphic kernel an a map is V$LT T - K : weak prespectra, natural up of to weak homotopy, cohomology of analysis Whitehead's related to closely is This result Further, R-prespectrum. a weak is T if a weak equivalence is which for of stringent less terms in theories on spaces their and spectra of notions E + there is , .& E E LV$E p: weak equivalence a natural such spectra of prespectra Define by (coordinatized) weak have been using. maps than we composite following the map: identity the is that condition deleting the inclusion the on Definition weak define 1.7, in 8. 1 V$E. V$E A V$LV$E A by $. requiring the R-prespectra equivalences, and define weak be weak to is the Finally, a natural weak equivalence w:RLT -+ LQT such that there only (and Oi+l D maps of weak prespectra by requiring cri Bi 23 a r! = that diagrams following commute: Weak holds). equality when case the for map term retain the R-prespectra RK = UV$E RT ------+ Qv$LT and LV$RE . and spaces and on cohomology (additive) determine maps weak theories morphisms Two weak maps determine the same morphism if thereof. are weakly homotopic. Similarly, that ; we then say 0: 0 and 0' in = Hy 0. 11 0,0' E [Y, E] are weakly homotopic if 0V = 0'V in HZ' we say that two maps Together result standard representability arguments, this with implies V. for each indexing space we view In of Theorem 1.8, see that are 0' and 0 theory a cohomology to uniquely extends spaces on theory cohomology that a weakly homotopic and only if if is of Ei-'y. kernel the in the 0' - 0 liml C spaces on theories a to extends cohomology on spectra and that a morphism of 0 epimorphism E Y E1yi. - lim 4--

24 morphism of theories spectra but not, however, uniquely since cohomology on we are ring spectra, and IV shall construct E chapter always honest ring w I which weakly null homotopic HA and thus are there non-trivial maps are in Thus, chapter VII. spaces ring E spectra from our applies, it where in m 1 Formally, induce the of spaces. morphism cohomology trivial theories on lim terms. the periodic For for any need circumvent work will of analysis C -1 theories cohomology of on equivalent to the spaces and is category WHA and 0 = (BG) liml KU vanish-because terms K-theories, the relevant C theories functor the forgetful to corresponds cohomology from Hd - WH/9 space compact KO-'(BG) vector ZZ over for a finite dimensional is any 1 on theories cohomology to spectra on spaces. terms [14]. For the connective K-theories, the relevant lim group G Lie notion We also require the analogous result for products. Recall the shall these of track keep distinctions We by [ll]. Anderson of results vanish below. TI') IX.2.5 from prespectra weak of T -* signs (Our a pairing (T', of shall generally just given, arguments the of because but, in this section the right. we ) write suspension coordinates since [SO] Whitehead's from differ on of rest ignore them in the the book . It pairing determines a spectra of E + E'AE" 8: that a map see to easy is and fix We we and spectra, spaces periodic of to the study turn now Via (v$E', v$E") -, V$E of ~(63): prespectra (compare IV. 1.3 below). weak As a harmless discussion. integer d positive an even throughout the simpli- th proof theorem, previous the of can verify the the of two one lines either of assume we in lie all spectra fication, zero- spaces henceforward that the of following addendum XII]. [48, spaces CW-complexes. the category V of type of the based homotopy of T of weak prespectra (TI, - T") f: 3.4. Proposition A pairing For Definition 3.5. hl/ IIc WHL, , 6 or define = 6 the category = homo- weak $(f): LT1h LT1' -, LT of spectra, unique up to determines a map be (X.X) pairs of category the E to , where X of periodic objects in topy, such that f and KO ~($(f)) * (K, K) pairings are weakly homotopic (X,X) + (XI, x') n% is and x:X -+ an isomorphism in Ijf . The morphisms 5: d -+ then spectra, of a map is E - E'AE" 8: If LT. po$(w(8)) TI') (TI, is are maps the -rX1 5:X that Sox = ~'05. such to weakly homotopic p). (PA 0 63 space functor to The spectra periodic from zeroth' Proposition 3.6. pairing a notion to ring prespectrum, rise The notion of of gives weak categories. periodic spaces is an equivalence of work shall We Proof. in (as prespectra coordinatized with this of theories cohomology products in and for notion is adequate the study - one-point with each Definition only by HA 1.7) the taken B of dimension d and with tV the requiring a weak Define ring spectrum in on spaces [SO]. i d Let maps I: T. to Titl for cr. i, 0. all that so V, of compactification The proposition and to weak homotopy. associativity up to laws unit and hold = - Xitl + B%~ a,: let and n%,, x:Xi x. let X, = X. Let . IIhv 6 (X.x) ring imply that prespectrum T determines a ring a weak weak theorem and prespectrum a weak is LT. spectrum ] cr. {X., Then X. of adjoint the be 11 d Ha and morphisms cohomology of in The distinction between maps since d map {x.} weak prespectra (because, a weak of is (Xi}: {Xi} - Q is 1 d the theories spectra on spaces ring between weak distinction concomitant and Q homo- of is Q% even, the interchange of coordinates self homeomorphism d Ew The in introduced be to spectra ring and spectra are ring folklore. KX Cl -r KX z: define to and 1 e = L{x~) KX Define the identity). to Lopic

25 ~lll~ll I I 1 11 45 be L(x.3 and the natural isomorphism of the composite valid ). WH.~ for clearly remain Hd for ) -c (E 0 , 5 0 ) Clearly 00: (D 0 ,6 0 d w-': Theorem of L{X.} 3.3. -R ISd{xi} Given a spectrum periodic connective maps, the thus With we evident of equivalence an is periodic spaces. IIhv -+ IIWHL . K: obtain a functor first By and the the naturality of K a map 6), the last part of the previous proof applies verbatim to (D, yield th d of Theorem K:{x~] 3.3, + map v$L{xi} specifies the zeroth diagram of -+ y the of inspection By . WHS y: D zero KDo such that R y 0 6 = 600 in an of periodic equivalence spaces (KX, + X) (X, an induces a space level, -r.y is y isomorphism for i 2 0. Therefore Conversely, ). given 1 and ), define yo = define l:E - IIWH X and the & composite of the between isomorphism natural identity functor (E, write 6) IIwH.2 , , (X,X) = (E E 0 0 yi: Ei = ER~~ as - inductively X composite the The conclusion follows formally from Proposition 3.6. 1 P d and I chapter in will been has already consequence following The used as {yi)o{~i)=hi)o{yi3 isaweakmapand +{X$ 2:vjdE {yi Then deeply more be used chapter in VIII. be weak maps. Define y: E + KX the composite of pl: E to + LvjdE and By the naturality of p and Theorem 3.3, second diagram of the ~{y.). be 6') (Dl, and 6) (D, Let 3.9. Corollary spectra periodic connective (E, y: ) spectra periodic of equivalence a weak is of (KX,~ -+ 5) course, (and is a Then there A: periodic of a map be 6') spaces. (Dl, -+ ) ,6 (D and let 0 0 0 0 th are isomorphisms weak equivalences in several times, used have we as unique map d: zero- with spectra connective periodic of 6') (Dl, - 6) (D, ). WH~ a A of equivalence weak . spectra. is A equivalence, then an is A If map con- periodic in not really interested are spectra but in "periodic We the given is A Lemma 2.10 by weak to (up map unique as Proof. and nective spectra", write we the for ,dc spectra. connective of category : such in commutes diagram following WH that the homotopy) / periodic of the category LC, IIWH Define 3.7. connective Definition spectra, to be the category of (D, pairs 6), D is a connective spectrum where d d and 6: D a map in is D R -+ equivalence an is Do R - Do tjO: that such & WH are of The morphisms 5 : (D, 6) -+ 6') spaces. the maps 5: D -+ D* such (Dl, and need be 6; A Note that, when is an equivalence, only one of d in . 5.6 = 6'oX WHR. R that The the periodic corollary characterizes an to be equivalence. assumed Proposition 3.8. The associated connective spectrum from functor also We a multi- need spectrum associated to a periodic space. connective to spectra periodic categories. of an is spectra connective periodic equivalence plicative elaboration applicable to periodic ring spaces. let 8: D -+ E be its (E, 5), Given spectrum a periodic Proof. - is ring space a basepoint together with X A a space Definition 3.10. 2.11 spectrum of associated Lemma connective and note 2.10 Lemma that unit point and 1, products @ and 0 such inverse map additive an and , @ d d a unique map gives D 6: + D SZ 50 0 = R 6 00 such that results these (since (so @ zero for a strict is 0 ring axioms and to homotopy up hold that the

26 IIRh'V, homotopy is diagram X) E following the (X, Proof. Given that factors @ through is to is X); X XA be commutative if @ said homotopy commutative (where b determines x and T denotes the transposition): as an H-space I. equivalent to X X 4.61, n X [26, X is commutative. By 0 0 under O-component. denotes the the Define IIRh2/, 83, where Xo category subcategory of periodic ring spaces, to be the of llh'\/ are objects whose pairs (X, X X) such that is is S2% - a commutative ring and space x:X adjoint to the composite Q, XAS~ - X lAb XAX 4 morphisms the (XI, - X) (X, 5: x') the 0); s~(X~, = -II~X E are [b] some for defines a pairing + {Xi}) ({Xi}, {Xi} X = = X - XAX @:XihX. Therefore X' of ring spaces such that 5*[b] = [b']. maps L:X - that Note x is auto- J itj Proposition by KX spectrum ring a weak determines and thus structure on matically an H-map with respect to 83 and is determined by its restriction of a map ring is X K -+ O:X K spaces. that implies result also That 3.4. 0 n nOx: nOX - T~S~% = components X; and by basepo"t to S2tx0 - Xo since d Conversely, given E IIRWH~ , it is ) straightforward to verify that the (E,5 X E X-module w free the be must generated r rOX, [a] for [a][b] = (n0x)[a] 0 d the $E {yi): v + {xi), Xi = Eo, in proof of carries 3.6 Proposition map [bl. by Therefore constructed. to that just by the product induced on E the pairing WH~ = % For or Definition 3.11. = the IIRG, define WHdc, a map spectra. weak : of ring is KEO + y E in category of periodic ring objects to , of subcategory the be n whose Proposition to complement analogous The is 3.8 a direct conse- is a weak commutative ring spectrum the objects pairs (E, 5) such are E that d valid of ring spectra) quence for (which clearly remains 2.11 weak Lemma is the composite to adjoint E -. 5: E S2 and IA er ) E IIRwH~~. b d last the of the argument just given to (D, 6 of application and part EAS -EhEO --+ E for some [b] E @ constructed in where Lemma 2.9; the morphisms as rdEO, is Proposition 3.13. associated spectrum functor from The connective of weak ring such that spectra maps the are E' - E 5: 5: 6') 6) - (El, (E, periodic an equivalence ring connective is to spectra ring periodic spectra 5*[b] = [bl]. By Lemmas 2.4 and 2.5, the zeroth space functor & -t hT1 categories. of induces a functor IIR - IIRh?/ . 6) Corollary (Dl, connective be periodic 6') and (D, Let 3.14. let ring spectra and liO) be 0 periodic ring -, 6') 0 (Do, h: a map of (Dl, have the We Proposition complement 3.6. to following (D, con- of 6') (Dl, -C 6) periodic A: map a unique is there Then spaces. th 3.12. Proposition functor zero- space from The periodic ring spectra to ring nective with spectra map zeroht X. of periodic ring spaces is an equivalence categories.

27 Proof. KA in the proof of Corollary 3.9 are maps The and y - 111. Orientation theory ring from Lemma 2.10 that A weak is maps and it of spectra, follows well. as so Remarks 3. be Let the connective periodic ring spectrum (D, 6) 15. coho- notion respect to The an extraordinary with orientability of Do that determined by ring [b] of a map is adDo. - D : (J Suppose E 0 mology use the here shall theory We theory. bundle to central is coordinate- Then such that spaces the ring .rroDo. JI*[b] = n[b], where n is a unit of the to theory orientation relate to I1 chapter of spectra free geometric of 6; adjoint the A*- of think a monoid-valued We I. chapter of spaces classifying shall maps which G functor bundles sphere of theory specifying a as F to i [b]) map yields a corollary the [b], = an - (J*( Since equivalence. is etc.). (orthogonal, topological, we 1, section In use the general shall n th (J. spectra ring connective periodic of 6) (D, -c 6') (D, 3: map with zero of theory in [47] to make rigorous a folklore treatment developed fibrations for GV-bundles oriented with respect to a commutative orientation theory of construction bar two-sided the use we shall 2, In section E. spectrum ring Ii2 to give a precise geometric description of a classifying discussed in space for E-oriented GV-bundles. That it does indeed classify B(GV;E) be more from the much will general classification theorems for deduced and bundles with additional structure established in fibrations [47], and be cther consequences of the general theorems there will also several discussed. orientation theory. Elementary 1. is It theory of Thom complexes and orientations folklore that the smoothly if one starts with spherical fibrations particularly works D * X with a given cross-section cr:X -+ D such that cr is a cofibration. : TC; to be the quotient space D/X. For Thom complex One then defines the of an the idea is to think bundle, the n-sphere bundle obtained by n-plane applying one-point compactification to each fibre. The Thom complex ob- tained in this way will usually agree with that obtained by one-point compacti-

28 replaced In smash fibrewise by pro- are joins fibrewise context, this taking the total fication space and will always agree with that obtained by of with Y and X over bundles GW and GV are $ ducts. Explicitly, if f and bundle quotient the unit disc of by its boundary (n-1)-sphere the bundle. spaces total Q Y G(V the be to f~$ define D$, and DE W)-bundle XX over Clearly, if the homotopy type of of the TS is to be an invariant where the equivalence identifies D$)/(M). X (D5 = $) D(~A with total space equivalence homotopy fibre of class must f, then the latter notion be defined 1 wedge point the (ox, $- y)v(f-'x, uy) to the (ox, Y. X X E y) (x, each for uy) terms preserving section of in maps fibrewise (and homotopies). if turn, In and section from the is $ fx from induced is induced The projection A JI f homotopy equivalent fibrations induce to fibre X1 + X are homotopic maps because it by an F(VQ W)-bundle is eh$ is in fact a G(VBW)-bundle a. X u from be formulated in terms also 6, then the covering homotopy property must by and it because and 5.61 [47, a reduction from those of 6 $ [47, inherits maps. fibrewise preserving section of It how obvious immediately then not is use 5.6 and 10.41 and of the Whitney sums given in I. 2. 2 on the bar construc- much the example, for goes through; fibrations of of the standard theory usual We have an evident homeomorphism tions which appear in [47,10.4]. a spherical fibration for procedure replacing a spherical quasifibration by L~(fn$) T~AT$ . fails. clearly G(VQ W)-bundle to be the Q 6 $ = X Whitney sum Y, When the define of $1-31, the basic theory [47, fibrations is redeveloped with fibres In induced from 6 A $ by X the diagonal map A: X. We then X + X over X preassigned category lie fibres and constrained spaces of to any in maps of 3. homeomorphism have a be V Let the be to 9 take and space product inner real a finite dimensional r C J of subcategory spaces of the (based) homotopy type of consists of the which - $1 Q T(E ~f *($)/T$ * $ by 6 :DC + X; induced from the GW-bundle is f*($) where over Df and equivalences. homotopy (based) their tV The basepoints are fibres of course, of induced X over GW-bundle the Df from $ f *($) by u:X + is [47, required to define which are cross-sections fibrewise cofibrations (see and inclusion cr used map GW-bundle the induces the 1)' = cr 6 over (since This condition both allows our proposed construction of Thom com- 5.21). quotient to define the on the right. plexes and quasifibrations with is necessary to circumvent the problem definition and recall the & be E a commutative ring spectrum Let E [47, $51. mentioned above shall We call spherical fibrations of the sort just * of E X from 11.3.1. "FV-bundles". specified GV-bundle if E-orientable 6 is A 1.1. Definition there exists a class Now assume given a grouplike monoid-valued j*-functor G together of a generator to restricts n free the p that such tV, dim = p En~f, E -+ F of monoid-valued J. *-functors, as in I. 2.5. morphism with a G n*E-module E*TX fibre x of 6 as fibres are thought of for each (where Define a to of be an GV-bundle together its with a reduction FV-bundle the of points over GV-bundles space). base structural to in The monoid this GV. precise meaning of a "reduction" Clearly 2. Let fl:D-+ E be 1. a map of commutative ring spectra. Remarks generality is specified in [47,10.4], GV- and the cited definition shows that Conversely, if connective and is D D-orientable. is it if E-orientable is 6 with bundles equivalent to Steenrod fibre GV group are bundlss naturally tV. effectively on tV when G acts group-valued and GV is and fibre

29 ii.8 then is D-orientable if it is 5 i,O, for is an isomorphism E-orientable morphism n : p " n*E) ; E*(TE+ -+ T*E) G*(X+; (because E~T~ 8*:D T5 -+ (n-1)-connected). is an isomorphism since TE is and theory depends only on con- By it follows 2.12, 2.11 that orientation II. determined by the preassigned R-orientation of 5. is HiioE-orientable if itis E-orientable. 5 and that a spectra nective bundle only ensure Of course, the finite-dimensionality of X serves to R for Henceforward, write iigEo. ii E the is (HR)*(x) that Recall = sequences. spectral the of convergence 0 orientation reduced cohomology E*(x,R) . By an (or ordinary R-orientation basic other remarks summarize following The facts orientations; about ; ~"(TS E p a class understand we 5, a GV-bundle of Z) = such R) if R and the previous theorem, are immediate from the definitions, the proofs en a generator of (TX; R) for each free H restricts p that R-module the to above. facts the about Thom complexes recorded an said (5, p) is then GV-bundle. R-oriented to be the x ; pair fibre Since Remarks X Let 1.5. be (finite-dimensional) CW-complexes. Y and with a definite fundamental to R"(Tx; n*E) restricts E*TX, p we identify can tV. X'A = X E = EV:X tV &X satisfies TE trivial (i) GV-bundle The class in each fibre x . E~T~ for an The suspension of 1 6 EOX' is under E-orientation of E ; it is image Definition 1.3. An E-orientation of an R-oriented GV-bundle €, is a )r called the canonical orientation and is denoted 0' E~TE, p c. class p to fundamental class tV, dim = n restricts that such the is v) (+, (ii) If a map, is Y f:X+ Y and over GV-bundle E-oriented an be the X; each fibre for E~T~ pair (5, p) is then said E-oriented an to of Tf: is T+ -) is ~f*+ the an where £*(+), of E-orientation (Tf)*(v) then GV-bundle. then the is f further, If, cofibration, a map induced of Thom complexes. Thus E-orientations are to be consistent required with preassigned with cup product v induces an isomorphism R-orientations. The following isomorphism theorem Thom the of proof IX)) E*(Y/x) 5 E*(T+/T(JI definition. this precise should help motivate Let denote the X of X' union the (by lemma). five the and sequences cohomology exact long and a disjoint basepoint. (iii) E-oriented are v) (+, and &) , (5 Y, X and If over GW-bundles and GV be a finite .4. Let (5 ,p) Theorem1 an E-oriented GV-bundle over where Y, )*4 then v) is an E-oriented G(V B) W)-bundle over X X (E,-,+, dimensional CW-complex X. p defines iso- an the cup product with Then v under the p~ v is the [48,XII] product external image [email protected] of t * E*T~. E*X+-C morphism Therefore E*T~ free E X -module the is p. by generated 4 Proof. determined cup product The is by reduceddiagonal the When Y, = X denotes ) v A (TA)~(~ E-orientation induced the - p B) V t Tfj - X A (which is induced via Tc A 1 from the ordinary diagonal 5 t Dc the total space). Now up induces a morphism of DE - DE+ t~ (5 E-oriented -bundles W) B) G(V and GW are ) (iv) If (JI, v) and w B) +, . . spectral the sequences which, on Atiyah-Hirzebruch E -level, is the iso- under over X, where 5 is a GV-bundle over X, then the image p of 1 2

30 54 composite isomorphism the B(SFE, = E) B(SGV; and *) *). SGV, B(GV; GV, = B(FE, E) f . w = v )rB) that Such of is the unique E-orientation in following the have then We which commutative "orientation diagram", so over -bundles X, and GV equivalent stably are J, and 5 If (v) GW inclusions evident the are is defined to be R) B(GV; and i the maps (WB) J,B) to equivalent is 2, 2) €([email protected] Z) for e @ f that then 6 is some *): GV, B(FR, only if J, is E-orientable. E-orientable if and e T 9 B(SGV; E) - BSGV > SGV . SFE GV-bundles 2. Classification of E-oriented the previous section all of the notations and assume that retain We spaces sight are in the category in of spaces of the homotopy type &'of in are X and G, Y, if in h' X) G, B(Y, 61, [46,~. By CW-complexes. X. is of SGV denote the component Let the identity element of GV. q can be quasifibration sequences are by rows The maps [47,7.9], and the = roE0 FR denote the group of units of the ring R and let Let of universal determinant thought as bundles. of the the denote Eo C corresponding components. union FE Define We shall interpret natural transforma- functors and the geometrically FE d: = FE rr -. SFE FR to be denote the discretization map. Let FE C spaces maps the of and by the 0 hM/ represented category on the homotopy tions the When R. is E to the identity element of the component corresponding that Recall brackets [47J. of orientation appropriate quoting results diagram by 0 0 .-, in par- space the Eo = QS coincides with F; sphere spectrum Q S , in based paper denoted that but denote classes homotopy unbased homotopy m and SF. = SFE F = FE ticular, In the we may take the general case, here. classes 4 e: Q SO -. E to be an honest map in A rather than just a map in H,4 unit first note by [47, 9.81, We that, , BFV] is naturally isomorphic [X m II.Z.7), also write e for the composite and (by we Next, by [47,11.1 X. to of classes of equivalence FV-bundles the over set 4 FE, GV -+ FV c F R~. c v 5 and 10.41, to the *)] is naturally isomorphic set FV, B(GV\FV, , [X of definition, the a homeomorphism have we a spectrum, of $1, 11.1 By of the structural with a reduction X over FV-bundles classes equivalence of ,.d EV) F(tV, cr: Eo -. for each finite-dimensional sub inner product spat; V Here, by equivalent homotopy is *) FV, B(GV\FV, [47,8.9], GV. monoid to V, sub- of We restrict attention to such with a and we identify FE to BGV and the maps ?. We are monoids given a morphism of space of F(tV, EV) via and Bj: BGV " BFV B(GV\FV, FV, *) + BFV q: -. GV GV right action defines a maps of and composition of tV), F(tV, C FV F. group-valued is G if - course, Of be j: G interchangeably, used can

31 and on then BGV also classifies Steenrod fibre effectively acts GV tV, [47,11.2]. forgetful B(GV; E) + BGV obvious Therefore q induces the q: group bundles X fibre over and GV with tV [47,9.10]: here the map Bj GV-bundles to GV-bundles. E-oriented from transformation under- the transformation its bundle to a fibre induces sending obtained by that admit of topological monoid such a structure If FE happens to GV- from fibration, q whereas lying spherical induces the transformation FE/GV is of - FE monoids, homotopy is so = E) B(GV; that a map e: GV t structural to of reduction the the forgetting by FV-bundles obtained bundles fibre the to equivalent the is BFE] , [X E w*(Be) then BFE, - BGV Be: of BHV classifying Given i: H + G, monoid. can be regarded as FV- either only by E-orientability obstruction to the of the GV-bundle classified bundles or with a reduction of GV-bundles structural their monoid to HV. = E are examples only the I know, as far As BGV. - S = E and HR cr:X be con- obstruction will a similar Nevertheless, below). discussed (both Theorem 2.1. X For [x', B(GV; E)] A' naturally isomorphic , E is in structed much more the next chapter. generally over GV-bundles E-oriented of classes of set the to equivalence X under equivalence. the of preserving GV-bundle relation orientation Example 2.2. Let E = HR. It is not hard to construct for a model HR Rather than give the FV-bundle E: D - X An orientation = F. p of an Proof. First, let G monoids. of is +FHR a morphism e: such that GV of maps a homotopy class described as can be note we that Bd: details, R) a homotopy equivalence, is B(GV; HR) B(GV; for that, any EV such * D map + the com- : fibre, some equivalence into homotopy a based is which D tV rows and bottom the middle use of thus we can and is, since -+ FR d: FHR the basepoints Here FE. in lies EV -+. p+:tV determine the fibres of posite de: orientation the GV diagram interchangeably. Clearly 4 FR is a morphism B(de), and B(de) factors cross-section of factors p a based g, through because is Te map px and the of fibre to equivalent is R) B(GV; monoids, of -1 px that also FE E ensures The condition D. C (x) X: inclusion for each in contained we is BSGV and have R), B(GV; BaoGV. - Bd: BGV through n * ri*E. over TX E of a generator is In the restriction of p E E Te to E~T~ the commutative diagram: following [47: defined with an FE-structure is p 10.61, the language of and 10.1,10. 2, K(aOGV, Bd > 1) A BGV BSGV is F = EV). for result the Therefore G (FE, admissible pair the to respect [47,11.1]. of a special case general an orientation of a GV-bundle For G, only on the underlying FV-bundle (and not on the reduction), hence depends BGV is only and if R-orientable if cr:X by classified GV-bundle The the for result for G by [47,11.3]. Alternatively, F implies the result aoGV if Z2 and either or (1) = 0 GV .rr if holds this and 0, = w*~(de) = when G is group-valued and GV effectively on acts tV, appeal we could 1 first Z2) (X; H = wi class Stiefel-Whitney is the or 2 = R char a*Bd E fibration- [47,11.4] rather than to the quoted to the bundle-theoretic result if a homotopy equivalence is R) B(Gv; BSGV zero. the By diagram, theoretic results. a GV = FR. of E-oriented universal explicit give The proofs 11. and [47,11.1 41 0 classified 13) with base B(GV; E) and with a by GV-bundles (a,

32 58 t GV-bundle the (integrally) oriented be €I0) (v0, by Let classified is of GV-bundle of classes homotopy set the to isomorphic (iii) ,GV] [X t 2). B(GV; in BSGV of inclusion the cr*(no. Via is + ff QO), [X , BSGV] - + gives tV XX adjoint its GV, tV X a: Indeed, given EV. + EV equivalences by SGV-bundles, of classes equivalence of set the naturally isomorphic to a has image in of GV-bundle map v the second coordinate corresponding the a. understand we other In canonical orientation. with a GV-bundles which if Tv only and if SGV V. & of orientation canonical the preserves ff us allows Theorem 2.1 words, preferred to orientations choose compatible induce diagram the orientation e of The maps (iv) the transformations * all their structural of reduction simultaneously on with a GV-bundles monoid the is po where send map v to which the E-orientation (Tv) a GV-bundle SGV. to EV. of canonical E-orientation 2.3. Definition GV- E-oriented An E-oriented SGV-bundle is an is There trivializations that and orientations between analogy an preassigned bundle and an that is R-orientation SGV-bundle that the such morphisms Suppose given important role in the an applications. plays induced from the canonical orientation. G F H A group-valued. of H where *-functors, $ monoid-valued is map The forgetful evident the R) induces B(GV; + B(GV; E) Bd: By 11.11 (and, for fails cases HV where [47,10.3 such as H and Spin = is transformation from E-oriented an R-oriented GV-bundles and isomorph- to t , to tV, [47,10.4 and 11.3]), [X effectively on GV/HV] is naturally act iso- ism on components. cogni- takes E-orientation Our original definition of an HV-bundles X. over GV-trivialized morphic to of equivalence classes the of set of zance fact that the set the equivalence classes of E-oriented GV-bundles of induces -+ B(HV; = *) HV, B(FE, *) HV, B(GV, = GV/W Be: E) Remarks 2.6. under is the union of the inverse images X (Bd)* of the elements of the over * transformation the is (Tb) (5, to b) (5, pO where ()I&), the sends which X. The image of i Bd. under B(SGV;E) of over GV-bundles R-oriented set diagram following The GV-bundle. the trivial of E-orientation canonical precisely BSGV R). B(GV; is C This implies the following corollary. commutes: + 2.4. [X , iso- X For , Corollary B(SGV; E)] is naturally 6 the X. to morphic over SGV-bundles E-oriented of classes equivalence of set We analysis of the complete the two rows of the orientation upper remarks following in the diagram (compare [47,11. 21). : EV -c EV v In upper row, the T the transformation which sends induces t 2.5. Remarks [X (i) to the isomorphic is ,FE] the of E-orientations of set to and ) v EV, transformation. evident forgetful the induces q ( cr: Indeed, XXtV adjoint its FE, + X +EV given X. over EV GV-bundle trivial Since so F/Gv. = E) E(GV; F and S, = = FE that Let E 7. Example 2.. . p gives the corresponding E-orientation and only if in SFE has image cr if cr BGV Be: a GV-bundle sends which the transformation induces BF -+ to 5 of fibre each of class canonical fundamental to the pa restricts EV. homotopy fibre stable its to the obstruction is class this class, equivalence transformations induce the diagram orientation the of maps-r The (ii) This since, seen directly be also can if fact S-orientability the of 5. E-orientation an send which the P). EV, ( pair to EV p the of class equivalence of

33 V and Sm, an S-orientation p: TE - asm has adjoint a map Rm = tV = induce B(GW; B(W; $1: B(GV; E) E) x . B(G(V t w); E) - large (if the base n of Sn 5 is compact), and for - sntm suitably X DE . 5 of is latter map the a stable trivialization of second coordinate the GV X over GW-bundles and Y E-oriented and Let (e, p) and (4, v ) be (EA by is v) pn 4, classified Then by classified and F. ; B(w smash Finally, we relate Theorem 2. 1 to fibrewise F). X $)(; products and f the observation that on based is proof The f imply 11 11. and 5.6,7.4. [47; discuss its naturality in E. 1: can be covered by a map of E-oriented G(V t W)-bundles from $) B( that w ; Remarks orthogonal and V Let 2.8. be W finite inner sub dimensional and E) B(GV; bundles over the fibrewise smash product of the universal f product spaces of is of definition Rw. It an easy consequence the the of to the universal bundle over B(G(V t W);E). B(GW;E) E determines maps given HA inBE3] that the product on smash product on f : By 5 spectra. Remarks of commutative ring be 2.9. Let E1 - E a map $:FE EVAEW + FE and PI: X FE on an appropriate (depending W) t E(V + I w actions y, the following 1.1, ,IL spectra of a map of and definition the of the such linear isometry CB RW .r RW) R that following diagram is the t commutative: are diagrams commutative: - >F(~v,Ev) and FEXGV IX1 >FEIXGV J F(tV, L > FE1 E'V) @ F E t >F(t(V t W)) W),E(V E) B(GV: BL: a map induces I Therefore following the and , El) B(GV; * required in possible 8 ; product on E is (at an honest map least if the given by I obviously commutative: diagram is formal inverses of weak equivalences would mildly complicate the argument f {Vi] and {w.] are expanding sequences such that Vil Wi to follow). If B(GV;E) > BGV f FE > Rm a sequence = of maps (0:) t is there then (UWi), and i i compatible with $:FE X FE + FE. The dia- fli:~vih EWi W.) E(V. t so -. l l on F and of the right actions y by the GV gram and the definitions of w GV-bundles of E and E' oriented the universal construction The 8) and (T, following diagram also commutes: on FE imply that the I complexes Thom of that the map shows [47,11. ] 1 in GI1) (TI, by induced diagram that the such is BG $ w and since products, with construction commutes bar the Therefore,

34 spectraY ring E IV. is BG induces the transformation commutative. homotopy We conclude that P3 t~) GV-bundle El-oriented to the (5, which GV-bundle E-oriented sends an for the cohomology operation it also where we written & Gp), (5, have determines. previous chapter, the basics In orientation theory were the of for theory represented by a commutative ring developed a cohomology obstruction In to analyze the E. to orientability, and spectrum order many other purposes, it is desirable to have a more structured for of notion a ring spectrum. C Eo and the consider the unit space FE To see what is wanted, The product $: EAE+ E SEE the identity element of r0E. of component Recall - E determine H-space structures on FE and SEE. e: S and unit when E that, S, FE = F and SEE = SF. When E = kO, Adams pointed = out [4, $71 that the group of in spherical fibrations over X kO-oriented to obstruction the that and J(X) the analysis of role a key play to ought in Now the be to the d-invariant. ought related directly to kO-orientability BO = SFkO, -+ induced from the H-map e: SF d-invariant can be thought of as 63 were to admit a delooping $61 and if e Sullivan pointed out in [72, that B(SF; + BBO then the fibre of Be ought to be equivalent to kO) BSF Be: €3' and Be therefore ought to be universal obstruction to kO-orientability. the Thus one wants at least suffiaient structure on E to ensure that One's first thought is to FE delooping). admits space (or a classifying monoid. One cannot re- of topological insist that admit a structure FE quire @ be associative and unital, to passage to homotopy, since without the smash product of spectra is itself only associative and unital after passage homotopy. However, one can ignore the smash product, re- to a pair- of Whitehead's notion of a ring spectrum defined in terms to vert * (by J.P. Ray) F. Quinn, and N. May,

35 i \ 11) 1 I I I I Illl~ll1l! ll!l i I 65 64 of theories E-oriented G-bundles and corresponding new cobordism theories. of [80], and assume that the given pairing is strictly associative ing spectra these Can questions. work many unanswered raises new Our and been used has satisfactory in theory, perfectly is This by and unital. 1 calculated ( theories be in interesting cases ? What is the relationship an investigation the orientability Stong bundles Patterson in of and of [52], E and spectrum E m ring an theory determined by cohomology betweenthe I only known the since practice in little use of present at is but strictly FE? of that determined by the T*E What are the implications for the composition pairing) and the (under S spectra ring associative are I that is here we show that I existence of an E ring structure on E? All a ring. R is where HR, Eilenberg-Mac Lane spectra m theories cohomology the interesting of most of spectra representing the 1. section in spectrum ring shall an E of notion the define We m I have an enormously richer been studied structure than has yet internal E is such a spectrum, FE and SFE will not only have deloopings, When I this potentially powerful of I and that to a collection gives rise structure Paradoxically, the although will be infinite loop spaces. actually they thee-ries. new are thus much structure than ring the stronger im- an of implications E m I ring strict it of not hard to construct E is plications associativity, m I other examples, we shall Indeed, among in section 2 that the see spectra. Em ring prespectm and spectra I 1. including as such various Thom ones MG, spectra exotic MTop and MF, is space to to display One way infinite loop prove that a space is an first In VII, will author the as they occur in nature. ring spectra E are m an action of an E operad on it as [vI,! We think of spectra 11. already m spectra from spaces ring a machine for the construction of give E m multiplicative additive we and structures, shall superimpose carrying Em This machine will construct appropriate internal structure. with in defined Since spectra, as actions structures operads. of by means by K-theories (geo- ring spectra which represent the various connective on indexed product inner sub a1 finite-dimensicn the spaces are 11,l.l' metric and algebraic) and the ordinary cohomology theories. 2 isornetries operad linear the give 1.2 to is it R~, of natural I. of In section 3, we study E-orientation theory when E is an E ring m the examples would it in this chapter, For a privileged in our theory. role takes E-orientability to the obstruction Here a conceptual on spectrum. it but , 7 essential for allow to chapters later is suffice to use only "Stiefel-Whitney" class inthecohomology theory is it form: one a degree an E assume given general operads. Thus we more operad fj and a m give a number shall We of the infinite loop space determined FE. by By abuse, we shall think . elements of -+ $ operads morphism /!! of this obstruction how show in to examples calculated favorable cases. can be given to map the via linear isometries as (j). 2 of (j) for B(G; E) E-oriented stable G-bundles the classifying space Moreover, think We chosen basepoints. to refer operads by Actions actions of itself an is the Thom space, infinite loop and (G = O,U,F, Top, etc.) spaces denoted is on relevant basepoint the multiplicative, and h as by spectrum GV-bundles, E-oriented M(G; E) associated to the universal spaces, not want to but we on structures additive impose by 1. We do have cohomology new Thus we ring spectrum. V E an again is R~, C m let do to impose zeroes. Thus want 3, denote the category of spaces

36 0 -space + X where SO = {0,1}. By a fi S cofibrations e: together X with 5.(~(g; hl hk)) > " '' 8 ,$)(V1 ... Ty(g;hl, ATV. ... TV1" V.1, @ .. . J J or with zero, -space such that (X.5) understand a /5 we O-space, x.)= 0 0; any inotherwords, x. re and Ej(g,xl = ,..., if Xs J to (j) half-smash equivariant the through factor b required is X -+ xJ X Cj: (j)Xz. XA.. .A x/A (j)~~ * defined with respect to the basepoint 0. b Let j J of b -spaces with zero. [ye] denote the catego=y 0). = jr if (0) (or V. B ... B - where W f...+jr J ) . . hr(Vjl+. - r +jr-l 1 we The spaces TV of a prespectrum have given basepoints, which de- (b) * TV is the identity map. (1): TV all taken to are below with smash and wedges be 0; products used note by then the following diagram is commutative: Z (j) fJ s g If and T s (c) j' respect to these basepoints. for By a unit T we understand A% E E for or jf) s Sj(g7) WO in (and, resulting that the A in E 4 So such a map e: >3 S 4 T Q @ or e: Tg7(V1 V.) . . . 8 '83 > . . TV. h A TV~ . w J J by 11. 1 SO + To or SO - Eo determining) a cofibration. and map .J 11.1.7, is and pe Let and prespectra with .Be denote the categories spectra units of units). with units (and morphisms which preserve The constructions and re- sults of $1 extend immediately to the categories LI . , ge, and Be 'e of definitions advised to review the is reader The of 1.2), (VI. operads is in ranges g as g Ej the continuous through For W, and V. fixed (d) actions by operads (VI. of 1.3), of the linear isometries operad (I. 1.2), and /3 which (j) consists of those elements such that of subspace the to proceeding before follo~ng 1 (II. definition. .l) spectra prespectra and W. g(v1 8 ... 8 V.) = J Definition 1 5) -prespectrum u, b A .l. a is (T, unital prespectrum LW and commutative:' diagram following the V. i (j) i' then is (e) If g E together with maps u) (T, U .A UA.. T(V1 W.) . A.. Wl) + AT(V. > + ATV.A~W . . TV1~tW1~.. @ . . . V.) d Tg(Vl + TV. j J J J . . . A TV1 g.(g): J J J I 8 for j the as interpreted be to is 2 where G~,(Rw), Vi (j), and t 0, g cO(*) 0 e: inclusion S are conditions such satisfied. the following that , To -. then If g E )%(k), the r &Gr) for 1 5 r 5 k, and j = jl + . . . + jk, hr (a) is commutative. diagram following $* (where is a sphere-valued functor on t as in I1 9 1).

37 then (l), TV + TgV is a homeomorphism in the homotopy tl(g): A - .AX 8 + A is defined for product [48,X1], functor a smash In (f) E If g g ' w class morphism ) is E f *(R obtainable by every and T(~~v), ) restric- (2); all such functors become equivalent in the stable each element g E 3 be E (1). ((f) could & deleted; see Remarks 11.1.10.) tion from some g for each Our definition ensures that, & -spectrum homotopy . H 8 cagegory E and each g E 15 (2), there is which map E A in E - E Lf a well-defined (T,u, if (T, !d a is u) is a spectrum. A morphism 5) -spectrum g . E a structure of gives ring spectrum in HA commutative (T, +: 5') u', of (TI, + 5) u, is (TI, 0-1) -, u) (T, +: fi -prespectra a morphism of with Whitehead's a comparison theory, Although irrelevant to our diagrams commutative: are unital prespectra such that the following illuminating. notion of a ring spectrum may be Sjk) . . A A TVl TV. . . . . B) Tg(V1 r . QV.) J J RW Let &(2). E g let and -prespectrum Let 1.3. Remarks be a 6) cr, (T, {el, and {ei) bases orthonormal have RW B) Rm and tV that Assume 1. e'! 11 J. 5;(g) t . T1V1 T'V. A h . . B)V.) S- T1g(V1 83.. . J = Ti let V and of compactification one-point the is and TR' J Consider the following diagram for any A T. = ZT. u: = u. tRe. Titl. the its denote de] [ kj let and -prespectra 4 of category denote [pel /J Let + 1 1 1 1+1 is the obvious linear isometry 2 + Rge' 0 and 920, where d: Rgel p full subcategory of ] + denote !d -spectra. Let v : & [gel inclusion fJ [, the q+l p+l g(~P B) any linear isometry: RqS1) + RPfqtl is f: and functor. )j. E ring prespectrum (or spectrum) is Definition An -pre- 1.2. a w spectrum (or E -spectrum) over any of & with a given morphism operad w operads !j . 2 + have not do defined and not between We a morphism of notion need any different Em ring spectra over operads. an as determining underlying space, the u) (T, Think a prespectrum of wedge over all V E d of the spaces TV. Then conditions (a), (b), and *(RW) to 5. give this space are precisely the algebraic identities required for the (c) J of bo-space. Condition (d) describes how to a structure the topo- weave in II I u UA logy of should add that we only how how to make effective use of h , but we TAT ---+ ~ht(0)h~~tRe s+l P 9 P 9fl last the when the V. and W are all (0). The two topology conditions relate the to the internal structure of (T, u). 5. practice, (f) is used to define In J condition (e). lies definition in the of hrce the and T(~~v), the maps The left rectangle commutes trivially, the in trapezoids commute by (e ) two

38 - 0 0 two left triangles commute while and the the definition of a b-prespectrum, With .. h S . AS = So for eakh g E on 1 = E.(g) (j), Proof. 3 commute triangle homotopy by the the remaining rectangle very definition and zero e:SO-+x a morphism of /d -spaces such that witn -space h is a is SO diagram looks This a prespectrum. of (see just like Whitehead's M. 2.5), ex- T 0 conclusion The by taking X = follows with sero [ye]. & E X all for we cept that haven't mentioned signs. in point that, to The is pairing get a previous the applying lemma. and must Whitehead's use fixed chosen sense, f isometries we -. . g(~p~~q) q. RPf q' P + Be restricts to a functor nW: Qe Lemma 1.6. The functor in component may lie the opposite If f=f in our diagram, then f p, p*l,'q s+l isomorphism a natural is there and [Ae], -C [[email protected],I h nm: from f Rq) q'l. space of linear isornetries g([email protected] the - Ref in (led) q+l P* and the linear isometries requires course, our theory Of no such choices, By 11.1.4, (nm~)(v) = lim S~~T(V + W) , WlV. Proof. Let * of changes such all in the definition of a prespectrum efficiently keep track Wi map and define a Wi) + h 5) u, [@,I, (T, g E E b (j), and f. E S2 T(Vi of coordinates. f by commutativity of diagram the .) gj(g)(flh . . . A it ring and spaces E ring spectra, but E is interest in lies The J m m relationships that the show next We be- prespectra occur in nature. which tween the categories and e, @ give 8, derived in 11, § 1 restrict to ' e [I 93 [@ and [ be]. similar relationships between the categories Clearly we have forgetful functors h [@ -. fj [I and ,&[ A,] * [ ye] 1 To). 5 (T, by objects on defined ) -+ (To, 5 I u, * W.) + T(V. T(Vl A . + .. h Wl) htW ... tW1h JJ j . - to a functor zm: functor The 1.4. Lemma ye @ restricts A obtained by . (S2m~)g([email protected] ^ (Q~T)(V~) is . A.. (SZm~)(V,) Ej(g): .OVj) g,] -+ &J a natural isomorphism [%I, and there is [ zm: m these maps make verified easily are passage to and to limits, T i2 a = { 0) in our diagram, we see that the map h -spectrum. With Wi b and [Qel(zmx, TI, x E h [gel T E 93 [ Ql. T,) h [ ,=l(x, m -prespectra, rest the and 15 v'i2 - T of a morphism is 11.1.5 : I. of T Proof. E (X, 5) E i% [gell g For b(j), xi€ X, and V.E tVi, define is clear. 5 ..Ax.Av.) = 6 .(g)(xlh.. .hx.)h(tg)(vlh. . .~v.). j(g)(~l~~l~. J J JJ J previous The result. two lemmas imply the following a structure of h zmx -prespectrurn, and gives remaining the 5 Then mm0 S 1.7. S=Q Z: Lemma is &-spectrum, and e:S+E a (i(g):~ + X must be a homeomorphism. ) are verifications trivial. (For (f), each kf of a morphism is E. !$-spectrum every -spectra for 0 m is 1.5. z0S0 is a !j Lemma -prespectrum, and e: Z S T m w and composed. be can 1.6 and , 1.4 Lemmas X S2 = Qm a morphism of b T. Id every tor -prespectra -prespectrum - le restricts to a functor Lemma 1.8. The functor Qm:

39 de], Q gel -+ h [ [ and there is a natural isomorphism h m: and 2. Thom spectra *-prefunctors it often is x-space to an construct 1, As explained in I§ ! an $*-functor. construct simplest to first Analogously, to construct The needed following immediate consequence of this lemma will be an is it Y)-prespectrum, often simplest to J*-pre- first construct an in X) VII. Recall that CX = (Q w 0' [ functor. 1) in . p 1.9. The monad Lemma (Q a monad to restricts An j*-prefunctor (T,w , is a continuous e) 2.1. Definition for c E gives - m0 in )1 [ map 3.1; naturd E b [ Le], the Eo and con- associative, commutative, together with a T: functor 3 -' 0 a in structure of C2-algebra b 1. 7 [ ) tinuous natural transformation T.83 w : T X T + (of functors $* + Q*x t + T such that e: natural transformation and a continuous the "E is Eo that implies lemma an inVII, see shall we As ring m (B TW TV through TW. factors W) A T(V w : TV X + is which space", a space two E with (a) space structures so interrelated m l~e W) adjoint LT(V TVATW - (B tW has TVh composite The that satisfy the laws up to space distributivity the underlying H- structures (b) with closed of image and coincides with the identity map an inclusion all possible higher coherence homotopies. f see that we shall Moreover, a connective = W TV {o). when reconstructed (up to homotopy) from -spectrum can E be b 1 is commutative. W) T(V 83 W) 5 83 t(V diagram The Em ring space E the (c) 0' I 1. I 1.8 examples. Recall that any Lemma gives the following class of TVhTW tVA tW - I E space is an infinite loop space (VII.2.l)and any grouplike E space is an m a, Q: A morphism (T,w, + (TI, o', el) of J*-prefunctors a continuous is e) infinite loop space (VII. 2). 3. me. that = e' and Qw = @) wl(QX natural transformation T Q: -C Ti such Example 1.10. any For -space (X, b without zero, construct a E), t with *-functor X tW + t(V 83 W) the projection is also an w: tV The + (X ,5) with zero by -space 0 to X and h adjoining a disjoint basepoint T * t e: that asserts (c) identity. Condition the e with is J*-prefunctor extending a and evident the in -spectrum, 93 the is Qmxf then 5 way. a morphism of *-prefunctors. I +. X E any is 13' If A-spaces. of a morphism inclusion of xs in CM m 2. 2. Lemma (T, w, e) naturally determines An $*-prefunctor operad and fi = fi via , then any ha-space is a -space the h' X (T, u, x-prespectrum 5). an projection while hl, -+ b the projection be used in allows k to fi +;f) sub that ensures its restriction to The continuity of T Proof. - theory. Therefore present the is an Q X' E E spectrum any for ring m. m m m m ~nduces De- a functor h$*(~ ) -C hd. R inner product spaces of X. space fine u = w(1~e): TVA tW + T(V + W) for orthogonal pairs (V, W) of sub- m u) that (T, ensure w Then (a)-(c) of the associativity and . spaces R of

40 cn 0 of prespectra. a morphism is B S - T e: is and a prespectrum that Vi C Rm and g E y(j), define For - (also x-space an determines F T 1.9, I, and I. 1.6 in As explained denoted l~...~~.) T~(V -- ATV. ... s~(~):Tv~A .I 3 .., c comparison by FT) by passage of trivial A Rm. to v limits over the composite to be 'U that FT = M sub an x-space. There are evident definitions shows as .(By) T(glV,@. . ATV. . % T(v~Q.. .(Bv.) TV~A.. 'U T~(v~Q.. (BV.). . 3 3 J of Q*-functors FT and SET FT which give rise to the sub 7-spaces u, straightforward an then is 6) '2-prespectrum. (T, that verify to is It FM SFM and of Mo. Thom Recall *-prefunctors. -! as spectra next display the We a is us by there Schulte, and to Kochman, Becker, As pointed out bar an as the discussion of the two-sided construction geometric and certain of Lemmas 2.2 1.6 to J*-functors which leads via of class t 2. I. and 2. I. from .!,-functor 2.1 the E ring spectra Q X of Example 1.10. 03 m A. of Let Construction 5. G + F be a morphism monoid-valued 2. Example 3. 2. Abelian topological an be X Let with product monoid @ *-functors. Then left G acts from the by evaluation on the J*-functor e) and unit q. Define an d*-prefunctor (TX, w, by on and and left the from any t the be Y right Let *. trivial d *-functor f:VAV', for and (TX)(V)=xt~tV (TX)(f)=l~tf the right. The which map acts on 4*-functor from G with w maps the by given e and grouplike a quasi-fibration G is if is *) GV, B(YV, + GV, B(YV, p: tV) W) Q +x+A~(v (XXX)+~~VA~W $+,-i:~+ht~A~+ht~ and f t a GV-bundle and p if admits a cross-section u is G group-valued. q fit fit^ -cX htV. hl:tVg fibre has and tV + * GV-spaces of induced morphism the from t ~(1). case the special as recovered is that Note Moreover, T : tV + B(YV, GV, tV) over the basepoint of B(YV, GV, *). 0 The derived by t determined 7-spectrum is S . Q The 03 *-prefunctor *-functors. Define d of morphisms all are i and u, an p, 0 < -space structure on the zeroth space QS coincides with the q-space (T(G; w, by e) Y), N F structure derived from the *-functor $ defined in I. 2.5. This phe- GV,*), ~v)/B(YV, GV, B(YV, = T(G;Y)(V) $*-prefunctors. arbitrary to generalizes nomenon . from induced e the and w with Whitney sum of B(Y, G, T from and t) Definition 2.4. $ write and *-prefunctor. M an be e) w, Let (T, 7-prespectrum Y) M(G; write and associated Write T(G; Y) for the m for the application of by derived 3-spectrum the functor 0 the to m an In M and T sense, evident for the derived 4-spectrum 0 T(G; Y). r., 7-prespectrum T. associated FT *-functor $ by an Define are functorial on pairs (G, and TG = Y). Abbreviate T(G; *) ,., rU . nVTV = and FTf FTV = nf-'Tf for f:V -V' MG and prespectrum Thom the TG and called are M(G; MG. = *) with by the composite w: ?TV X FTW + ?T (V (B W) given spectrum of G.

41 0 RmS . with Clearly coincides Me When is group-valued, G G when full generality, In bundles. normal stable their on "Y-structure" MG define well as just could we sphere use by bundles the associated of F, define a G-normal space to be a normal space in the sense of to maps bundles principal the to fibration the [63,1.1] with a reduction of structural monoid of its spherical G(V GV X ~)/e B) GV. X V)/GV 83 G(V of G-normal then [47,10.4]; cobordism the gives n*M(G;Y) groups to G An intrinsic with a spaces their on fibrations. spherical "Y-structure" their total spaces (because However, since these bundles are not universal requires an bundle or fibration theoretic interpretation of a Y-structure preferable it seems use not are to case classical the in even contractible), general appropriate classification theorem, and results of this nature are construction. bar the G GI-structure a GI, to maps GI-triviali- [47,$ll]. When a is given in G When F, = fibration appropriate an by replaced be must p H is eation [47,10.3]. When H\G-structure maps to G, an a reduction obtain a fibration (spherical FV-bundle universal in order + BFV to DFV as in Definition of structural monoid to H [47,10.4]. When Y the FM is = the we Here cross-section). with new DFV/BFV; by replace TFV could of via G over *-functor composition maps. as regarded is and 2.4 right z an TFV would again determine d*-prefunctor be would TFV the old and the of retracts deformation new which define deformations via ones tV) + F (tV, TV) X FMV, = TV) (tV, F + F(tV, FMV GV X parameter value. A *-prefunctors for similar re- each $ of morphisms 1115 5 below. Remarks and 2 3. M-orientation a Y-structure is an by case the general mark to applies F. maps to T(G; Y) when G y-prespectra : TG vMG -. v Q~TG I, = Note that the map of In view I. 2.5, all of the usual cobordism theories except for of L defines explicit MG-orientations GV- the universal of MGV + TGV : represented by thus are theory PL While y-spectra. possible it may be and TFV DFV/BFV when G = F). Thus of equivalence the (via bundle hoc triangulation the manner based on to handle MPL in MSPL and an ad any reflects following The MG-orientation. lemma admits GV-bundle an certainly pre- is it , 2 5 I theorem, as BPL handled in and BSPL were H\G-structure the fact that (or G-manifold) with an a G-normal space bundle of a general framework these within treat to ferable axiomatic is, (that an FMH-structure MH-orientation). admits an - by [64]. in will be given author the second treatment Such a theory. H + G + F be morphisms of monoid-valued Let 2.6. Lemma Y our theory does other note one We to which important example FMH Then right is J *- of there + H\G j: a morphism &-functors. w the Brown-Peterson namely not yet apply, spectrum. point is that The such that functors over G morphism j coincides withthe + G F given concrete geometric a good requires theory our a not model, merely homo- diagram following the e and = H when commutes: topy theoretical construction, and no such model is presently for BP. known Y general in Construction 2.5, For treatment of the Lashof's [36] implies that Pontryagin-Thom construction G is group-valued and maps if the cobordism gives .rr*M(G;Y) then 0, G-manifolds with a groups of to SFMH. G maps to F or If then H\G maps to FMH or SF

42 Proof. H, and we have the commutative diagram B(*, = \G H G) Let G F be rnorphisms of + Q*-functors. The -+ H monoid-valued - commute because they following diagrams already do of level the on so B(*, HV, x tvr tv) B(*,Hv,~v) + THV , x B(*,Hv, GV GV) Q *-prefunctors: HV\ = .$MH MSH and where A is defined j: GV F(tV, THV) map. the evaluation is the adjoint composite. top of the to be MSH) M(SG; As in [66], the will be discussed and interpreted geometrically maps J-homomorphisms. bordism induce the FMH + H\G j: B(*, = G H\ G, r and q € maps upper contractible and the is G, G) H When in natural maps of J*-prefunctors a number the We record of and q on t3 maps HA in isorriorphisms prespectrum the (because the are remarks. following morphisms The same letters will be derived used the for in that, conclude We each V). weak level are equivalences for homotopy 7 -prespectra and r-spectra. of interpretations cobordism The should 1 (via MSG or MG ) E- Mj. off factor a direct q split the lower HA, maps above. discussion the be clear from lower of the such that the map E is identity. restriction to this factor the For (G, Y) as the morphism 2.5, in of Construction 2.7 (i) Remarks J*-functors G, t) -+ B(*, G, t) induces a morphisrn of $ o-p+efunctors B(Y, q: E ring spectra for Orientation 3. theory m Y) T(G; q: -+ TG. consider and with a given to map operad Let )j be an E of notation the (ii) morphism induce a Bh In previous proof, the the maps m the for BX shall write We 15 -spaces via this map. as x-spaces of H, t) which in turn induces €. *-functors = E(Bh): B(H\G, G, t) + Q B(*, a is This of first a grouplike de-looping (~11,530r [46]). X 93-space of a morphism : E *-prefunctors % TH. -. H\G) T(G; infinite loop an BX is equivalent as of notation abuse harmless since G If evaluation the then *-prefunctor, d an is T and F maps to if (iii) a topo- be X happens to if of the usual classifying space to space X maps which F(tV, TV) X tV + TV induce maps E : B(FMV, GV, tV) -+ TV Pf -spaces (by VII,3.6). logical in of monoid category the turn induce in a morphism $*-prefunctors. T(G,FM) -+ 8 : of T is E - a e:S By Lemma 1.7, a E -spectrum. b Let be of the previous lemma and remarks give considerable The maps and thus h-spectra e:F FE of + is a morphism morphism of MG). M(G; of structure the about information the commutative ring v0E. h-spaces. Let R denote Remarks 2.8. from an For an x-spectrum M derived *-prefunctor, be $* monoid-valued of -functors, *F Let a morphism j: G write (I. &-spaces I. 1.6). 1.9 and j of map also denote the derived and let Qm~(G; M) FM) M(G; = and Q~T(SG; M(SG; M) = SFM).

43 This diagram is therefore equivalent to the diagram obtained by Write e any for GV G A F ~FE, VC Rm. - the composites of functor to the zeroth space a diagram of connective of application . 2 fi 111 orientation diagram constructed in Consider the features It fiberings (that is, which are equivalent in the in spectra the rows are for GV- B(GV; E) = B(FE, GV, *) E-oriented space the classifying fibration sequences to category stable of On [48,XI). spaces, the level bundles. Here FE as is via the $EV of subspace a identified in direc- both infinitely extends now sequence stable E-orientation the nVEv ;;! Eo : - homeomorphism compo- by is a right GV-space and tions: sition maps. of of a spectrum, the By definition 11~1.1, the following QB(G:E) - ... BFE-BB(G;E)-+... -G~FELB(G:E)-%BG-% if V and W Rm: of orthogonal subspaces are diagram commutative is ,., u - . si"+w~(v+w) H - Given G, we also have the infinite sequence Eo of map and a this the E-orientation sequence of H sequence into identification the Therefore subspace FE as of a of is nVEv con- V when R~). 8 = -maps (because the maps of are 2.6 111. GR~ = from V varies, and sistent inherits a right action by G as FE Be: of fibre to the equivalent E) is B(G; Since BFE, BG by action FE the Moreover, GV. the actions the right FE is X G - is B(GV; equivalent to the E) fibre shows that an easy diagram chase b-spaces. itself a morphism of Indeed, a comparison of (e) of we which composite, following the of Be: by denote again with I. 1.10 1.1 Definition follows that the cancellation from this shows BFE. -+ BG BF Be' BGV , (x~-')(~~) = xy for g:V -V', nVEv E x E and y Now FV. = nVtv by 5 be a GV-bundle classified 3.2. Definition Let as construction bar geometric two-sided the of the discussion a recall X ' BGV. Define w(5: E) to be the element a*(Be) of the group a : [46,§3]). a is monoid Abelian an that too Recall from (or 2 5 I -space 93 E) 6 of class Stiefel-Whitney called the E-theory is w(g; [x', BFE]. X for -space any & and that the discretization d: 8 + ?r X of a fd -space 0 FE is the obstruction to the denotes also its If E-orientability. and is of map always a By passage to limits from the -spaces b [45,§3]. the by determined cohomology (reduced) spectrum theory represented m diagrams orientation R C V for result. , we conclude the following 1t of (X ). element FE by FE, then w(g; E) can be regarded as an and all maps The0re.m 3.1. All spaces are grouplike h-spaces Previously, the obstruction to E-orientability was studied by are diagram the stable orientation f$-maps in analysis the Atiyah-Hirzebruch spectral sequence. Larry Taylor of 1741 has given a number of results so obtainable, and we are indebted E to him for several helpful conversations. When E is an very ring m spectrum, these results are immediate consequences of the definition

44 83 82 obstruction and To see this, assume that X is a theory. ordinary each finite is X and degree is in finite T*BF Since finite dimensional CW-complex. integer positive some for 0 = a(Bj.a) dimensional,. Therefore a. BFE homotopic null + E):X w(5; is If the (n-1)-skeleton on an obstruction set we then xn-', have 6 is Thus, if = E) wn(aE; C E) awn(S; defined. E) is wn(S; if 0 then H?(x;T~-~E) is torsion free for n> 1, if R-orientable and Tn-l~~) w,(s; E) c H~(x; T~BFE) = $(x; if example, For E-orientable. is E MU = E spectrum a ring is if or E) E) of a null homotopy of w(5; existence on to xn. Clearly wl($; the into which MU maps (such as MO, MSO, KU, etc. ) and 2) (x; H~~~~ if 5 Stiefel-Whitney usual the is obstruction to the R-orientability of of E-orientable R-orientable. 1, it n2. if is free for torsion is 5 then is 0, n> For 2. 2. III. w(5; and only null will be E) if finite, is X homotopic When T = 1) SFE 71. = FE T (E~, E T = 0) rrn(E0, , at localizations its if all primes, a set T from at or of and away n n n n E[T-~] denote the localiza- and -component 1 by given is isomorphism where the from translation the primes, are null homotopic. Let E T Y over GW-bundle a be 4 Let 0-component. the to classified by away tion of E Then at and 0. from T. Suppose that G = 45 classified is If $A* Y $: Then BGW. and X + of the composite by a MSp-orientable. is hence structure, a symplectic admits = 0, w 1 5 that follows it is 5 5 MSp[l/2]-orientable is MSp- and thus that V and W X -c BG([email protected] W). Take w:BGV to be orthogonal sub- BGW m the Since same is if only and if orientable it (MS~)~-orientable. of R . Then spaces the diagram into which ring spectra statement holds for these and maps MSp in- BGV W) f BG(V a BGW X KO and XU, MG spectra clude and most of the interesting Thom I BG BGX BG of at a problem only generally is bundles vector the E-orientability prime 2. given by the (b is the product where is homotopy commutative, By the definition of w(5; E) and the third author's result [65] Since conclude we H-map, an structure. BFE b-space is -. BG Be: n is , S bundle any vector 2, 2. n over for (ej)*:-r 0 -c T MSp zero that (5 A E) *; w that (5; E) + w (*; E) if w is defined and contains n n n n n 3, E into which E-orientable 2 MSp is for any ring spectrum maps. n E) are defined. (a; w and E) wn(c; Clearly w natural in in X, is n n applications like this can be multiplied ad Clearly infinitum, XI -+ then is a map and if w (6; E) is defined, X if f: the sense that n * and computationally and satisfactory a conceptually context gives our is E) (f*~; w E if + E1 similarly, E). (5; f w contains defined and 8: n efficient analysis the for framework E-orientability. of is a morphism of E then w ring spectra, El) E). fJ*wn(e; contains (S; m stable the if Returning diagram, that note we orientation to course, the under Of El-orientability implies its 5 of E-orientability be derived Q I, by E = M happens to then, from an 2 *-prefunctor, § assumption that the much weaker 0 spectra ring of a morphism is FM is this diagram 2.4, Definition *-functor of and the of use in HA. .- 4 from of -spaces diagram a commutative by to passage derived

45 the In following remarks we construct an j*-prefunctor functors. Analogously, although we cannot spaces in general. E-oriented G-normal -spectra, the f(l show consistency our of T(G;FM) for a general like Thom a -spectrum E, we can construct f.1 of record certain morphisms prespectrum by E) direct appeal to T(G: Definition 1.1. two definitions of when E M(G;E) d*-prefunctor, an from derived is M(G;E) and discuss the structure of when MG maps E. to as E) T(G; k-spectrum a Define 3.3. follows. Construction *) T(G; E)(v) = B(FE, GV, ~v)/B(FE, GV, B(FE, q: -. induce tV) GV, B(*, tV) GV, The maps (i). 3.4 Remarks and, e:tV4T(G;E)(V) inducedfrom for f:V+V1, T(G;E)(f) is B(l,Gf,tf). a morphism and thus q: T(G; E) + TG a morphism -prespectra of T: tV + from B(FE, GV,tV). For V orthogonal to W, induced is &-spectra. of MG -c E) M(G; q: E)(V)A~W u: T(G; -+ T(G; E)(V +W) is induced from the composite B(GV, (5) HV, Be: the maps G, to maps H If tV) B(FE, HV, tV) 8 -prespectra and thus G) -+ T(H; E) of induce a morphism Te: T(H: tw GV, G(v+w), B(FE, GV, tv) x t(v+w)), = B(FE, tv x tw) B(l~i+"'t~(~~, G) of M(H; h-spectra. E) M(H; a morphism Me: where i: GV -*. G(v+w) is the natural inclusion. these With maps, 5x 1 X ----+- F(tV, EV) tV + EV X tV FE The evaluation maps (iii) is a unital prespectrum. the maps For g 6 (j), T(G;E) maps induce GV, induce E: B(FE, a morphism tV) EV which in turn [email protected])) . . A T(G;E)(v~) A T(G; E)(v.) -. T(G:E)(~(v~ tj(g): . . a+. 1.6, &:T(G;E) -t vE of %-prespectra and thus, by Lemma a J morphism -spectra. E: M(G; E) E of )2j composites from are induced the *-prefunctor 3.5. Let E = M be derived from an T. Remarks the natural in- Mo + S2v~~ and maps P~TV 1% = Mo Then which in a tV) GV, B(FM, -. tV) GV, B(FMV, induce turn duce maps 2.5 Construction of and from b-prespectra T(G:FM) of morphism of M) T(G; to 2.2 Lemma In Construction used limits the of 3.3. view identifica- an is hJ-spectra map of induced the am, in the definition of T(G; E) well-defined and The verification that these maps give are tion. The maps E Remarks 2.7 in and and given the q on M(G; M) a structure straightforward. &-prespectrum quite tedious, but is of coincide. remarks previous T(SG; E) for the Write defined but with -prespectrum b similarly FE replaced by G and SFE. Write SG and E) and M(SG; E) M(G; %-spectra derived by application of the functor om. for the groups T*M(G; E) gives the cobordism 1115 [36 BY 1, [63 1, and 2, 0 of (normally) E-oriented G-manifolds when G maps to and of

46 \ Remarks Let y: MG - E be a morphism of .kj -spectra. 6. 3. following is commutative: diagram Then the theories k0-oriented bundle On V. lay to is this chapter of purpose of analysis an for foundations One the study bundle topological of and Sullivan's J(X) the groups of Adams' study purpose, this For of infinite loop point of the from theory view theory. space 2.8 M(G; that, in HA , MG -. E) q: Remarks from conclude We to understand it geo- on the depend work is essential their which portions of splits MG a direct factor off such that &-I) MyoMj 0 (via metry formal purely by follow portions which and representation theory) (and -c have We y. map given the to this factor on &:M(q;E) restricts E substantial that out turns It on the level. manipulations classifying space analogous by replaced G with result an SG. parts of their results can be obtained by elementary chases of a pair of large B(SF; kO) for kO-oriented diagrams the classifying space focusing on fibrations and on The functors and natural transformations spherical BTop. on CW-complexes by the spaces and maps in represented finite-dimensional are easily described, and it is simple to interpret the informa- these diagrams obtained the classifying space level tion in bundle-theoretic terms. on the construction take analysis of both diagrams, we shall For the and of o ore precise formulations the data will be given given. data following as ) later. (1) The Adams operations $r and their values on K0(sn) [i]. (2) of the Adams conjecture [2,17,57,73]. validity The The splitting of BO when localized at an odd prime [6,53]. (3) given the diagram, we shall also take as first For the kO-orientation of Spin bundles and The Atiyah-Bott-Shapiro (4) on ~s~in[i/r](~~) pr classes cannibalistic of the derived values [13,31. J(X) papers these results do not depend on Adams' last two Note that [4 and 51.

47 For the second diagram, we shall also take as given It looks just second main diagram. our construct we 6, section In that the k0[1/2]-orientation of STop fact The Sullivan bundles, the (5) one, like the first interpretation and its analysis is exactly the same; only the induced map F/~op[l/2] values the and equivalence, + BoQ9[1/2] an is its localizations Chases of give Sullivan's splittings of and BTop changes. KO[.t/2, l/r](Sn) on er of classes the derived cannibalistic These splittings, and chases, imply the odd primary [71,72]. odd primes. at TO~/O After for oriented bundles and theory stabilizing the classification 2, kO)(X) the prime from Away C2(SF; part [15]. calculations Brumfiel's of developed in fibrations a of an orientation by we what explain we [47], mean obstruction to the isomorphic is to JTop(X), the 6 invariant becomes the 1. theory in theory a cohomology to respect with bundle stable section We existence of a Top-structure on a stable spherical fibration, and the general diagrams which relate oriented construct bundle to certain theories JTop(X). yields Sullivan's 5 -invariant analysis of bundle theories in 2. cohomology section operations and to larger Very loop of this theory depends on the spaces. infinite of use little main section We in diagram our first construct 3. By its chasing the However, machinery developed book shows that all spaces in sight this in various spaces the of p in prime each at splittings we localizations, derive extra structure is essential to the applications. This are infinite loop spaces. section SF/Spin, and SF, in B(SF; kO), 4. diagram, such as These splittings, spherical fibra- kO-oriented fibrations, spherical for classes Characteristic Adams' [5]. in calculations of many imply chases, and splittings The of SF tions, terms in at present, described, be only topological bundles can of and (unpublished), role were noted by Sullivan F/O but the recognition of the and the infinite loop these operations are homology invariants of operations, and by following analysis the recognition that the this, from and, kO) B(SF; played important splittings Thus of which know to the it is space structure. at the prime 2 odd the to be new. appear primes at identical to that is formally spaces. are are here described only infinite loop and which of as homotopical of theorem main the of a version prove we 5, section In so and [4] is in our dia- displayed maps the The problem, then, of to determine which J(X). analogs theoretic introduce also We bundle the groups recalculate of grams maps and which parts are the diagrams commute on infinite loop E and 6 the gives 6 [5]. in Adams by invariants e and d studied of the In 7, section this of results combining by level. space infinite loop the obstruction to kO-orientability form stable spherical fibrations (and its of results of Adams and Priddy book recent with re- and still more [8] recent & kO). defined ring E the to IV53 is spectrum application on depends of sults of and Snaith, Madsen, [42] Tornehave and shall Ligaard [38], we m spherical stable kO-orientable of kO)(X) Q(SFi on over fibrations the group complete the infinite nearly our diagrams. of analysis loop in X and takes values a certain group to JSpin(X) JSp%(X). Its restriction of E-orientations theories bundle stable 1. is is JSpin(X) isomorphism. Therefore an kO)(X); Q(SF; of summand a direct notations those summarize we quickly convenience, reader's the For the group of j-oriented stable spherical fibra- the complementary summand is then establish notations for stable We in this chapter. be to used III from analysis This should be as a generali- regarded j. a certain spectrum tions for a theory mean an orientation of such by bundle theories and explain what we carried out by Adams [5] for the case when zation of that X i-sphere an is with with spectrum respect to a commutative ring E. i> 2.

48 91 ,'A I. be a morphism of grouplike monoid-valued j*-functors Let F j: G - CW- Henceforward, restrict attention to connected finite-dimensional be orthogonal finite-dimensional sub- V 2.5) (I. 1.8,2.1, and let and W and GV-bundles. 5 bundles GV' and GV and of spaces base as X complexes 8 classify GV-bundles and inner product spaces of R . BGV and B(GV;E) is equivalent 5 '8 EW1 to 5 EW 8 X equivalent if to said are over be stably 5 W) + BG(V + BGW X BGV o: and CW-complexes, over GV-bundles E-oriented =V8+W'. for some and W'_LV1 suchthat the WIV Write (5) for VtW and E) W); + B(G(V -+ E) B(GW; X E) B(GV; $): B(o, fibre- (external) induce the %G(x) Let {g} and call a stable G-bundle. of stable equivalence class 6 product and smash wise (internal) Whitney sum (III §l,2.1, 2.8). and Then BG, by classified is EG(x) of set the denote X. over G-bundles stable There is an explicit quasi-fibration sequence of the image and depends only BGV] in [x', BG] [Xt, on the dimension of V. The product on BG induces the external and internal operations and A Q FE GV , BGV E) B(GV; -> terns of on stable G-bundles described in the fibrewise smash product and which interpretation bundle-theoretic the is given in m. 2.5. of If i: H -r G 5 where $1, Q =I5 ($1 Q ($3 and Whitney (5)~ {+I sumby {5~+) = a morphism is of grouplike monoid-valued functors, is another there J*- bundles with V and GW . 1 W and + are representative GV explicit quasi-fibration sequence Let group Grothendieck G(n)- constructed from KG(X) denote the the Bi GV -L GV/W BHV - BGV the relations when = EV additional bundles for n 2 0 EV' (or, with m We have identifications C the R for V from GV-bundles dim V', = V dim ). and a map qBe + interpreted in III. 2.6, GV/W that B(HV;E), = q . such Be: The maps and q of the two quasi-fibration sequences above are defined in T {o}] [x', [x', = %G(x) BG KG(X). X = C [x', BG X Z] = BG] construction in and have analogous interpretations way terms of the bar same the under neglect of ~G(x) &(x) in denote of image the JG(X) Let of additional in terms transformations to and from bundles with structure, homo- (or, G-reduction equivalently fibre to passage group-valued, is G if hence the duplicative notation. there If G E -. E1 is a map of ring spectra, Thus topy equivalence). B(GV; a map BL: E) + B(Gv; El), interpreted in III. 2.9, such that qBt; q. = is + BF. Bj: BG C [x', BF], JG(X) = (B~)*[x+, BG] SG We instead of G when write all are given with a canonical inte- bundles ?(x) JO(X) J(X) for and JO(X) for his KE'(X); C Z X Adams writes with all E-orientations are required to be consistent orientation, and then gral has JO(X) no Z X geometric more is logical, but less convenient since notation the preassigned orientation. integral n*SF + n*SO j*: = J course, Of therefore uninteresting. is and structure ring Write G, G/H, to passage by obtained for the spaces E) B(G; BG, and s n: = n* stems is the stable denotes the J-homomorphism, where classical C R~. limits over V spaces (by I) and the first three are The infinite loop stable e., (i. spheres). of groups hornotopy 2.8) if infinite loop an is and space ULI. (by at least is last a grouplike H-space X over to said are p) and,GV'-bundles (5, and (E', p') E-oriented GV spectrum ring E (by an is IY. 3.1). E m pow) be stably 8 EW, p (B (5 is equivalent to (5'6) EW1,p'87pOW') equivalent if for some the and W'IV' such that V + W = V' t W'. Write {E,p) for W~V

49 { stable (5, p) and call of 5, p} an E-oriented stable equivalence class product formula E)(X) G-bundlk. set of E-oriented stable G-bundles the Let %(G; denote P(B) A $. (5 (5 = ~(g)} A b'(g)} $# A dg)} A ~(g)) (5, X. over image of [x', g(~; E)] E)(x) is classified by B (G; E), the B(GV; e (5) for all XY) E)(X 2(~; in ZG(Y). E ($1 and ~G(x) [x', E)] depends only on on product the and V, of dimension the B(G: in preserved fine the particularly interested in are We by structure internal A operations B(G; E) induces the external and on E-oriented B) and are equiva- which maps, infinite loop by which we understand maps in HT sum. product and Whitney smash stable G-bundles given by the fibrewise (see who reader The II,92). HA morphisms of maps zeroth to the lent in E)(X) can is defined, but be K(G; uninteresting. Grothendieck group A The discussion next section. does skip share not to the our interest may in E)(X) denote the image of Q(G; E)(X) of neglect under ~G(x) Z(G; Let based one two on proceed will there levels, on E-orientations and the other orientation. Thus based notion. following the on . -c B(G; q: BG E) = , BG] [x', C E)] B(G; q*[xf, Q(G,E)(X) An E l. 2. Let E-orientation an ring spectrum. E be Definition m BG g: H-map an is G B(G; E-orientation An 1.1. Definition E) -c of is an infinite loop map and if g E) g : BG -t B(G; if said to be perfect is map. homotopic to the identity is qg that such maps. infinite loop as = 1 qg Given + B(G; be E) denoted B (F; E) will again its Bj: with composite g, As will discussed in be § 7, there now exist homotopical proofs in diagram following the g and by will be homotopy commutative: somewhat composite cases of the important weaker assertion that the several com- B(G; + B(F; E) is an infinite loop map whose E) natural g and the of map posite with BG Bj: to map infinite loop an + equal is BF. e B(F;E) q: BF as shows (justified by [-La, I]) 8 in easy Barratt-Puppe sequence argument An only FE G admits a perfect E-orientation if and the if e: G + is trivial that In particular, G E)(X) E-orientation. an admits if Q(F; JG(X) C loop map. infinite gives an classifies E-oriented GV-bundles E) B(GV; that proof The w is G of of g E-orientation an When R C V over limit maps the sense, the 8); in an evident explicit universal E-oriented GV-bundle (T, In EV. -+ B(GV; gV: E), the gV induce maps TGV + BGV E) A T(GV; as V varies. (T, compatible are 0) } by classified be ~GX e (5 Let there- - vE and prespectra morphism TG maps these of define a practice, -c a:X as E), B(GV; through to homotopy) (up factors ga If BG. necessarily fore a morphism induce y: MG -.. level, of spectra. On the coordinatized E the element finite dimensionality by the V some of then X, for holds n appropriate compatibility of the statement that the , which destabilizes gR V and pro- (gn)*(0)] of E(G; E)(x) is independent of the choice of {(ga)*(n), g is an H-map, will ensure that is spectra ring in of a morphism y in } {e ZG(X). jects to we If so determined all for orientations p(g) write a is y that 11.3.4, explained in and after as sense therefore, and Whitehead's an H-map that ensures the validity of the then the g, by requirement g be The verification that y is HA. morphism of (weak) ring spectra in

50 I actually E a morphism of very and has not deeper spectra lies much ring is additive, @ and x on infinite loop spaces Since the stable category m yet been carried out in the interesting cases. Y, and Y + g: X maps f, two infinite loop maps. For an H-space are Conversely, let y ring spectra and : MG - E be a morphism of consider /g f of instead g - f write shall (We we when define @(f X xg)h. = f/g homotopy following the commutative diagram: an g and H-maps f if H-map f/g is are ) additive. as @ of think choose to Y and X If and homotopy commutative (as will always be the is Y case). are infinite loop spaces and f and g are an is f/g then maps, loop infinite infinite loop in map; particular, 1/1 A XX) the trivial infinite loop map. 6(1 = is BGV B GV abbreviate weak homotopy equivalence to equivalence. to agree We equivalence of infinite loop spaces, we understand the zeroth map of a By an Here j: -+ FMGV is in defined GV\GV IV. 2.6. E is a homotopy equiva- 1 - immediate The following result is an weak homotopy equivalence of spectra. lence (if GV E W ), and we define homo- gV = By* Bjo E for some chosen quasifibration sequence of consequence the inverse topy the -'. of & 2.6 with Comparison proof of [47, 11.11 IV. shows that gV classifies the E-oriented GV-bundle (T, p), where I( rr:DGV is the universal and BGV is its E-orientation induced GV-bundle Let 2.1. Lemma E-orientation the composite Then G. of an g be an define gV The MGV. -+ TGV : L via the MG-orientation from y x BG B(G: 2 E) B(G: E) E) B(G; FE x .-L%L- IV. 3.6 (with V, E-orientation g of G by passage to limits over and is g If H-spaces. of equivalence an is g) is an equivalence $(TX then perfect, diagram y ignored) k-structure from the shows of Thom how to recover infinite loop spaces. of of classifying diagram spaces. the displayed over spectra 2.6, By IV. main concern in this comparison the with is section of Our different + through FE factors G\G. e:G Since through is maps contractible G\G I() (6, Thus let E-orientations of the same underlying stable G-bundle. map infinite loop trivial and the is e 12.21, and by -spaces, X [45,9.9 of and F. and by classified X over G-bundles stable E-oriented be v) 15, &-spectra. g a morphism is y when of is a perfect E-orientation of G - q: since and BG BG :X qp l! qE homo- + commutes up to Since B(G;E) -+ X, and @ topywith classes Cannibalistic and the comparison diagram - q(Z/F) *. = = qe 2. @(~XX)A~F g(qaxXqSj)h r (.uxXSj)n null q, this T E) B(G; -+ is canonically equivalent to the fibre of FE : Since diagrams key commutative and some construct record We a few Clearly (presumably lemmas here. technical well-known) FE + :X that l! 76 such E/F. 6 a map determines homotopy Ot where {E~, ) 6) , E the unit 6 E (X We write in (multiplicative) H-spaces @ for the product on all sight, X a) A - z. Since 76 classifies @(~6 regarded X -+ SO is (with as an orientation of the trivial G(0)-bundle cO:X X Of course, X. inverse map and in spaces sight will have a homotopy H- all t groups. homotopy on inverse additive the and addition induce x and @ of Thom complex ), and since, by the explicit definition X 6 9 v in III. 1.5,

51 We note next that +/1 an essential role in versa1 theory. our level will play {E0,6) {5,6v v), @ {5,v] = @ v) = {[email protected] 5,6 c(+). factors also through conclude we that (5,~) = {5,6yv). homotopy commutative: The following diagram is Proposition 2.2. applications our In be construction, difference this of given a shall we space classifying of some for class Y bundles with and structure additional Y such The classifying qb. r qa that E) B(G; -+ we shall be given maps b: a, will Z' be ay and by and a classifying above map y :X -+ Y for maps ~d " a/b. Note that and 6 will be dy for a map d: Y -- FE such that If E is an E w B+ and FE on infinite are E) B(G; on + ring spectrum and also ~d homotopy , hence the d and ea/b, homotopy the null * q(a/b) " loop maps, then diagram on the level a corresponding diagram determines the homotopies are explicitly and canonically determined by the of spectra. of B+/l TO c(+) by the ~(+/l) c: c(+), and (B+/~)T r definition Proof. since B+o T = TO +. By construction, c(+) is natural in G. When G = e is so d. is H-maps, then are b and a If spectrum, E an is E If ring w Now + and 1 = on FE = B(e;E), hence +/l r c(+). T B+ = trivial, qb maps, infinite loop as and = and qa then maps, infinite loop are b a The last statement holds chase. diagram r T $/1 follows by an obvious c(+)o is and map infinite loop an d maps. ~d = a/b as infinite loop general observations above. by the The theory of cannibalistic classes fits nicely into this framework. Let ring spectra. + : E -. E be a map of B(G: -, E) B(G; B+ q, = qB+ E), Then interest main The theory in bundle but E-oriented an in not lies often Thus assume given morphisms and results a canonical H-map there FE that such c(+): B(G; E) -+ its relationship to a larger bundle theory. is B+ call c(+) We infinite loop map, then so is an c(+). TC(+) e if B+/l; H G F monoid-valued *-functors and assume given an of {g. p) is an E-oriented the universal cannibalistic class determined by +. If g write its composite with also We of H. g E-orientation for write c, by classified X over G-bundle stable E-orientations of G-trivialized There natural are now two B(H; E) -c B(G; E). Bi: maps the by given namely those sight, in H-bundles stable G/H& B(H; E). BH & B(H; E) and G/H Define on Z(G; E)(X) by +{$,p] + {5,+p) and note that +{5, is = by classified above shows that B+ i?. The discussion FE -+ an f:G/H where ~f, as factors gq/~e is quotient Their H-map +IS, 1.1 E E)(X). z(~: = u ~($1 (5, ~1 if perfect. is g is an H-map by virtue of Lemma and 2.1 infinite loop map result. following have we notations, these With the can define cannibalistic Of course, given an E-orientation g of G, we "comparison the following of three squares first The 15) = p{$] The p(g)]. c(+){e, = p{e] by for stable G-bundles p classes 2.3. Proposition these fact that classes are represented by the composite the c(+)g uni- on diagram" are homotopy commutative:

52 valid diagram homotopy commutes. These statements remain with H and G dimensional, explicit the HV and GV, V finite case in which replaced by E). B(GV; of H-space an of lack for fails f on construction structure the com- We shall need some observations concerning localimations of determines is a right the to g extends infinitely diagram the perfect, and If the integrally oriented to restrict We T. primes parison diagram at of a set of corresponding diagram spectra. on the level case in order to deal with connected spaces. q~ fact and the 111.2.6 that have we *, = Proof. By - and spectrum, ring a commutative again is ET The localization are do I not know if localizations of E a3 ring spectra SF(ET) e (SFE)T. but any infinite loop space information ring from the derived Em spectra, where the thus a is and of B(H;E) to injection the first map T takes FE genericall) X We under localization. preserved is structure Ew write commutative the have We ex. -- f~ 2.1. Therefore Lemma by factor at localization for T. diagram following For is any G, the composite a localization Lemma 2.4. 1 G When BSF[T- X (BSF)T r BSF F, = map the and ] in which contractible maps. Thus BiBe through is the infinite loop G/G is $ have we X) and commutes trivial infinite loop map and (since Bi with = A xBe) X $(gq (Bi) = ~f (Bi)~f is if E OD .is E an spaces infinite loop (of equivalence an ring spectrum). = (Bi)gq gq . In Proof. following homotopy commutative diagram, this of view the - third square homotopy orientation. an commutes by the definition of The is connected fibrations of the fact that localization from immediate preserves why wonder will sequences familiar with reader Barratt-Puppe The given by sign the writes appears. If one of down explicit equivalences x 9 Be BSFE - E) B(SG; BSG SG e SFE 7 and work- sequences, from fibration starting BG the two rows with honest two homotopy G equivalences ing left, one produces - QBG. turn These out Be I' IX - BSG BSFET - ET) B(SG; SG - to by differ g, given arguments Barratt-Puppe course, sequence X. Of a map produce determined, such not uniquely that the f, (e.g. 1521 ) [48, I I IX ET)T B(SG; BSFET - BSGT ----c - - SFET SGT commute. Conversely, homotopy squares left two Bf given such that the homotopy square commutes, there exists right g such that the rest of the

53 The hypotheses imply that f and f' induce the same homo- When G case we rename in F comparison diagram = the (in which Proof. - H = G), f Suppose ology. morphism on integral hom that are homotopic f1 and f intrinsically be can sometimes characterized in terms g. of obstruction the is T,Y) H~(x; e k If to the X. of the (n-l)-skeleton on g 2.5. be Let an Lemma of G ET-orientation assume that the and of restriction to the (n-2)-skeleton of a given homotopy, then extension the following two hold. conditions f*(x) - f;(x) = 0 for h = x H*X, E T-torsion. H,(SF/SG), H*(SFE), and r*(SFE) have no (i) respectively). are Q, H*(SF/SG) and and Z H,(SFE; Q) of finite type (over (the product the Kronecker is vnY + first @ nny) H~(x; : > , < where H~X (ii) square SFET that the second + is such sF/SG H-map unique the of the f: Then level calculation explicit chain by holding equality definitions). the from Since comparison diagram homotopy commutes. T*Y 0. = x> k, < a monomorphism, is H*Y + Since h: SF and through factors fl,f/f' Proof. such another Given H-map and T*Y have no (because Ex~(H~-~X, vnY) = 0 H*X is of finite type, H*X therefore induces the zero map on homotopy. = on homotopy. f; Thus f, by the 0 = k ZT-module), a is T*Y coefficient universal T-torsion, and Adams, of pair following the lemmas complete Frank by me to out pointed As Y when VIII. = proof a simpler for 1.1 RZ.) (See theorem. I proof. the me analog The by following out to pointed also was 2.7 Lemma of 2.6. connected homotopy associative Y and X Let Lemma be Frank Adams. type. finite of Q) H*(X; with H-spaces, H-maps If two induce Y - X , BSOT Y of X and spaces be of Let the homotopy type Lemma 2.8. the homomorphism same Q-+ ~c*Y 8 T*[email protected] then they induce the same Q, homotopic -f where T is any set of primes. Then two maps f,fl:X Y are + H,(Y; Q). H,(X; Q) homomorphism Q). H*(Y; * Q) H*(X; homomorphism same the induce they if Moore [50, the Hurewicz Appendix], Proof. By Milnor and homo- T(n) where BT(n), lim = SO(n), in a maximal torus is Let A Proof. morphism h: H*X tensoring induces + T+X a monomorphism upon Q, with diagram Consider the evident and be the inclusion. -t A let i: BSO Q) this and the monomorphism generates H*(X; image algebra. an as of a connnected CW-complex and be X Let 2.7. Lemma a be Y let follow- connected homotopy T-local H-space. that the associative Assume ing two conditions hold. ** - T*Y H,X, H*Y, and have no T-torsion. (i) Atiyah and Segal By monomorphisms. are KU(A), on and ch , Clearly i are and Z type finite of respectively). Q) H,(Y; (over Q, and H*X (ii) [14] limit argument, i* and an inverse is ch on a monomorphism. Thus N maps two Then f, + are Y same X fl: homo- induce the they if homotopic a KU(BS0) is a monomorphism. By Anderson [9, p. 381 or [14], c is also morphism -+ H,(Y; Q) . H,(X; Q) These statements remain true monomorphism (in fact an isomorphism).

54 after of representing spaces BO and BU at T. localization BSO and the KU-~(BG) = 0 and no here (since maps phantom are There Proof. - It that follows localized real K-theory f same map of and f' and induce the for G), it suffices to consider a compact Lie and KO-~(BG) is finite group * * [X, Y]. since fc BSO X BO B0(1), that f = (f') : therefore, [Y, Y] -+. of inverse the systems completed (real and complex) representation rings this valid, the same remains result In fact that the shall use we VIII, by n, where odd only consider well as may We of Spin(n) and SO(n). completions proof, for at T. why rational observation The following explains = . , . . P{kl, km) R(SO(Zm+l)) = 1)) + RO(So(2m of determines the information the self-maps respect behavior with to to and injectively R(SO(2m-1)) to surjectively maps T*BSO 2-torsion in T ' . . ,Am. Am} . C RO(Spin(2m+l)) = P{kl, R(~pin(2mi-1)) . 2.9. Lemma - BSOT T. s BSOT be a map, where 2 Let f: Let by a check R(Spin(2m-1)) characters, Since A m restricts to 2A m-1 in of a BSO E ZT be such that f*(x) = a.x for all x 6 a. ZT G . T 4j J J spinor the to contribution inverse no make representations the exceptional then # y E If a2BSOT, 0 f*(~) = a y . (i) 1 limit. If 0 # y E s~~+~BSO~, j 2 1 and k = 1 or k = 2, then f*(y) = a y . (ii) zj reduced statements, are understood to both the coefficients mod 2. In be Proof. let For (i), p (BSO; Z2) be the unique non-zero primi- H 6 kO-orientation Spin and the J-theory diagram of The 3. n n and recall that tive element know not exist I do if there involve main examples The K-theory. 1 2 1 * * p5=p4. Sq*p3=p2 'SQ*P~=P~ 2 and Ss and KU , w ring but explicit E ring KO Em spectra which represent if 0 = 0, and the displayed equations show f*(y) = Clearly if and only f*(pZ) represent and kU which kO the associated connective theories spectra By 0 = f*(p2) an obvious only argument with the if f*(p4) = and 0. if that are constructed in = BU SFkU and BO 63 - SFkO; informally, €3- Write VII. BSpin cover E BSpinT of a, f (p ) = * For 2. mod 0 0 a if and only if the 1-components these infinite loop spaces BUX of are Z and BOX Z. T * 4 1 that (ii), simply recall generates q if x: s8j -+ BSOT xo ir BSOT, then y = For G is equivalent as an = and G = U, BG* 0 3.1. Lemma 8j 2 tlk: q y map. the non-trivial is s8j + s8jtk = where , x 0 or BG(1) X BSG (8. infinite to space loop Anderson I learned from which result, following the Finally, Snaith, and BU(I) = 2) K(ZZ, 1) and K(Z, BO(1) admit infinite unique = Proof. - t=i BSpin and Y BSO when X true remains 2.8 implies that Lemma ' T T structures, space BG(1) by obtained + loop B% map natural the killing 2.10. Lemma induces BSO isomorphisms BSpin 4 T: natural map The or ,,w (i. e. we, map, and infinite loop automatically an 2 is ) c nl or T 11 on and complex K-theory. Therefore real [BSO, is + [BSpin, BSO] T*: BSO] define BSG @ and T: BSGQ9-t BC&, , as an infinite loop space and map, to be classifies (which @ BG isomorphism. an 4.5, By VI. the inclusion q : B G(l) natural fibre. its The composite the canonical line bundle) is an infinite loop map.

55 BG(I) BSG x B%-L- B%X B% T map which is non-trivial on @- the equivalence. desired is [13] have constructed a kO-orientation of Spin Atiyah, Bott, Shapiro and of spinc. kU-orientation a and Spin maps we if to well-defined G, Thus, have because =I 1 T I, isomorphisms on and i f~ induce ex composites both 5 T H-maps BSpin -. : kO) B(G; g and f: pin -. BO in [email protected] is null homotopic by the splitting and because the component of e €3 * diagram. = B(S0; kO). the comparison by ex f-r of SO, B(S0, precisely is the of Spin, fibre The *) quasi-fibration -c B(S0, *) and case, interesting) complex (less the in The argument fails I have i: BZ2 = ~0(1), and explicit equivalence gives an SO/Spin. -- BO(1) this rz that case. in q verified whether not not fi or we Similarly, if spinC maps have H-maps to G, to the We of study turn rest of spherical fibrations. kO-oriented The and f: pin' pin' BO - B(G; kO) - g: @' with the construction and analysis of the will be section the concerned i: an have and sO/spinc + BU(1) we equivilence explicit "J-theory diagram", which is obtained by superimposing diagrams involving 3.2. Proposition equivalences: following are The composites of an elaboration the classes on the Adams conjecture cannibalistic and diagram for comparison BSO X BSpin kO) k0) B(SO; X B(S0; kO) A B(SO; €3 and BSpin g: kO) SF/Spin and B(SF; - f: ' ""€3 X kU). B(S0; --?-I- kU) B(S0; X ku) BS~~~~=B(SO; BSU €3 f restricts and on we to that just discussed, Of course, this map SO/Spin of chases easy proven by is result The Proof. compari- the relevant following the have observation. son diagram of Proposition 2.3. facts the salient case, real the In that are the composite ~F/~pin - F/o and of natural map The 3.4. Lemma and that w2: 2) K(Z2, - BSO of BSpin equivalent to the fibre is are BO(1) :Bog+ w of equivalence components an the f and 1 [email protected] isomorphism -+ an is e*: k0) ; (6 T~SO (because the obstruction w 2 of infinite loop spaces. BO(1) X F/O -+ sF/~pin of IV, 3 5 be 2 homology). can non-zero mod on calculation direct by or In (SF;kO) write abuse, By QB(SF;kO). = G and RBS = 0 Define @ case, complex the that are facts salient the pin' is equivalent to the Y, + X a map 8: For QBG and etc. SF, Spin, = G when interchangeably or -. of a comparison by (equivalently fibrations) 3) K(Z, BSO w3: of fibre T (homotopy and L generically for the projection El3 - X from the write BSO X K(Z,2) + K(Z2, 2) and that T pin' = Z. [email protected] 1 + 1 €3 L : to the fibre of 2 be F8. -r to (~F:k~)/~pin Define RY inclusion the for theoretic) fibre and Corollary fi: sO/spin + BO(1) The composite 3.3; homo- Bog is the of This space classifies g). equivalent to is (which f of fibre the fibre BO topic to thenatural inclusion q : BO(1) - spherical stable kO-oriented as trivializations Spin-bundles with stable 63. Proof. diagram, where Consider the following any SO - is BO(1) : I, - a morphism from derived Qg: Spin were (SF; kO) Just if as fibrations.

56 106 of write monoid-valued $*-functors, kO)/Spin BSpin and T: (SF; kO) -. (SF; kO)/Spin -. q: (SF; and for T (obtained by Barratt-Puppe sequence arguments) for qa a map diagram &q a a L in the and on the following page. -irr such that TS~T notations, the solid arrow portion of With exists and is these this diagram commutative Proposition 2.3. homotopy by of diagram, the right At the defined (as an infinite loop BSp% is to the fibre of w2: [email protected]+ K(Z2, 2). space) be claim We with dotted arrows inserted, this J-theory diagram that, r, 2 and all spaces in sight is homotopy commutative when exists and To'see this, first recall the following calcula- localized away from r. are ) 2.9. also Lemma (See and 1. ,5.1 [l Adams of 5.2 tions such KO(X) i/lr is a natural ring homomorphism on, 5. Theorem 3. Jlr 5 = er on line bundles 5. Let x 0. %O(sl) = -ir.BO, i > 6 that If i = 4j, ' Jlrx = r2jx; if ie 1 or 2 mod 8 and r is odd, i/lrx = x. that i/lr determines a morphism of ring spectra 11.3.15 It follows by is defined away -. that c(+l): B(SF; kO) -+ BO k~[l/r] thus k~[l/r] and @ isomorphism from -iriSF -+ [email protected] is an Since for i = 1 (by Corollary r. e*: -iris translation i = 2 (by are from R~SO -+ BO, where the and for 3.3) by the smash and tensor product squares of the generators of generated the c(+~) a LO) is 2-connected. Therefore B(SF; lifts uniquely to a map, 's), 1 still denoted c(i/lr), into BC~ is defined to be the fibre of this map, [email protected] BO Cr defined localized be SZBC'. For r even, is a BSO BSpin to and away from r. For r odd, Theorem 3.5 implies that Jlr/lr Boa + Boa is bundle trivial (because the square of a line BO(1) is trivial) and annihi- on lates nZBSO The splitting SO a Spin X RpCO determined by the fibration Q9' 5: n non-trivial on is SO which and any map -- Rprn Spin sO/Spin + SO -) 1

57 r BSO + - : BSO shows that SO/Spin BSpin is null homotopic. q: + BO +r-l: Therefore BO a map to uniquely lifts BO + +r-l: BSpin. BO gr/l: Similarly, lifts B08 '1 lo ,r/l to uniquely a map br/l: [email protected]+ BSpin , and +r/l 2.2. Proposition by T c(+~) BSO Bs0,- 8 Q9 c(+~)~; = Define is cannibalistic class. pr thus the Adams-Bott pr diagram the is homotopy commutative by Lemma 2.8 0, = 6 If (or, if 1661. (See also Lemma 2.9. ) p. following calculations of Recall the Adams [3, 2 regardless T, Lemma 2.7) since, / of what 0 does, ~(+~-l) and 9 > If i 4j, 0. i -iri~Spin[l/r], E x Let 6. 3. Theorem = map rational homology by Lemma 2.6 and on same the induce (+r/l)f3 or +i(r2j-1)a = 8 mod 1 f rss for 1 = prx 8, x :if iI 1 prx 2 1 mod Con- r ~'(+~-l) (t/~~/l)p'. s, from away particular, In 3.5. Theorem 2 2j r and prx = 1 + 8. mod 3 _f I x for then diagram versely, if the is homotopy commutative, e* = e* on * on rational homology ( +r/l by the known and behavior of +r-l x Here translation fro& to ~*BSpin -c 1 Sx denotes the isomorphism q homology) and thus rational LL. by the cited lemmas. In particular, 9 below). BSO and BO for .rr*BSpi% similarly (and 4.a - - the of pr and ur are the r, 2 and away from 3-theory diagram maps are analyzed 13, in $21. The numbers a = (-l)jt1~j/2j e z[l/r] 2j 9 [3] Adams r, has constructed an H-map from Away homotopic. homotopic away null is BSF -%-+ spin A BO The composite * [email protected] which has the cannibalistic class pr as 2-connective pr: BSO LL. from [73], by Quillen [58], Sullivan r, or Becker and Gottlieb [17], since * from directly follows also a map such of existence the and rover, Therefore a reformulation this statement is just of the Adams conjecture. ur Clearly Lemma Adams' map and the 2.10. simply connected cover of in- BO pin such not uniquely is yr +I-l. qyrr that yr: exists there @ insist In we can and do particular, that its restriction to the determined. samd the duce homotopic. thus are homology and rational on homomorphism td ! shall retain the now duplicative we ur and For clarity, notations translate of 8 when BO(1) be the trivial map r non-trivial the and I mod 1 f d r. is but class homotopy its not most important about u what is pr, since to map when f I r S~/Spin C SO/Spin 3 mod 8. 9 r r remarks the That diagram above and location in the 3-theory diagram. its u Define fy : BO + BO = of fibres the below, 3.7 Remarks By d ' Q9 with compared which should be 2. result, following the give Proposition 2. by abuse, equivalent; BSpib -t BSpin pr: of and are . M~ by both denote we ' diagram commutative: homotopy is the following r, from Away 3.8. Proposition Define! of the J~ and J' to be fibres of +I-l: BO + BSpin and 8 g BSpin ; B (SF '? kO) sequence 4r/l B08 -t BSpin8. Standard Barratt-Puppe : arguments then give rrr rr rr , a! , $ maps 6 , and 1) and that such (QC(+~), ), (1,a y , ), y , $ (1, Er, (GC(+'), of completes the construction This fibrations. of maps are 1) the diagram. at of set Localize all spaces in sight primes any . 3.7 Remarks T and con- sider the following diagram, where any e is an H-map and e is map:

58 techniques, Adams and sequence spectral using Adams [8], Priddy Corollary TT~Bo[~/~]~ E i > 0. Let 3.9. x 4j, = i If proven thefollowing recently have as an infinite BSO of characterization ; or 2 mod 8, 1 I i if ..x 1)a - +L(r2j 1 = arx r mod urx for + = 1 e 8 1 2 23 space. loop and dx = 1 + x for r E + 3 mod 8. i Proof. is immediate from Theorem 3.6. For = 2 For i 2 4, this exists stable There the in isomorphism to up one and, 2. 4. Theorem -. Lemma from follows it 2.9. For choice of yr 1, and i it.holds = by our only one connective spectrum the zeroth space of category, which is equivalent Corollary 3.3. (or prime. given any at BSU) of of the to BSO (or completion) localization write again we v, p For odd, vl, w, and wL for infinite loop maps I X (p-locd) space loop infinite X equivalent which split X as W any W- for 4. diagram analysis of the J-theory Local splitting requisite The ) F/~op. BO to include BSO. and (Examples each analyze the localization of We the shall J-theory diagram at 63 We shall need the following of exists by the theorem and the splitting BSO. 3. = r(2) information. maximum yield to Let so r chosen with p, prime as the image of which in the let odd, p For be r(p) any chosen prime power immediate Lemmas of 2.6 and 2.7. consequence group is motivated units. This choice of r(p) its generates Z of ring > H-map between an be Y + 8:X let and 2 Lemma Let p 4.3. P by the following facts [3, 5 21. Let Z denote integers the of the localization homotopy BO. ti-spaces Then as type of same the (PI at p. and W-Y Qv~v&v: Y. v~~LQv':~+ QvLe In 4.1. Lemma (r2-1 , Z Let r(p). = r p=2 isaunitif 2j in basic result summarizes The following information contained 2 and or if p > 2j is 2 and mod (p-1), while 0 a unit if p > rZj-1 Theorems 3.5 and and 3.6, 4.1. Lemma , 3.9 Corollary (p-1). 0 Zj mod are p3: BSpin + spin and = BO 2, u3: BO -L p At 4.4. Theorem 63 @ Throughout this in spaces all specified, section, unless otherwise the > equivdences: are 2, composites following p At c quiyalences. p and r at are assumed to be localized write We r(p). denotes sight IL. 1 w'B0-L W'V BokBo~w 80 AW, BO, and BSO, BSpin interchangeably when odd. is Recall from Adams p 63 an as splits BO p, odd at that, [53] or 41 Lecture [6, loop Peterson infinite space W as = wL where niW = 0 X i , 2j(p-1) when r.W = Z unless 1 (PI' letter W is chosen as a reminder that W carries the Wu classes The ( WL ox *r-~) urx LBO 1 XBO-WXW, BOX BOA BO 63 * w. = Q-l~jQ(l) in H (BO; Z ), where cP is the canonical mod p Thom and J P vxv-' 1 W' vL for splitting maps from Write and v and isomorphism. W LBO wxw 'BOXBO mB0 XBO to @ 63 63. @ I BO and write w for and o1 W projections from BO to W and. .

59 ~llli!ll l)~)!l~i~ 113 . TRT Spin k0)/ (SF; - pullbacks, It unusual to encounter pullbacks (as is opposed to weak y X 7~75;17F in deleted) is the which uniqueness clause in the universal property for the s~/spin L s~/spin x s~/s~in L homotopy category. the However, the equivalences of theorem imply the following result. Corollary 4.5. in the homo- a is pullback At p, the following diagram equivalences: are composites following the 2, > p At 4. 8. Theorem category: topy k013 AB(SF; LO) B(SF; XBSpinXBO BC BCXWXW (i) Gr-1 P P - 09 - BSpin BO 2 (SF; nTXngXnT.. ko k013 (SF; ) (ii) XSpinxsO c xnwxnw'-Lc P P QD p KlTTx -.& (SF; kO)/Spin X (SF; (SF;ko)/Spin kO)/Spin --?-+ XM C (iii) P p Write X the for localization at p of any space X'(') which appears e in the J-theory diagram and write f of localization for the at p when fr(@) P f and one of the first five Greek letters. We thus have J M C BC is . P' P' P P' Corollary 4.6. equivalences: p, the following composites are At 6 (Y E BO and kO) B(SF: on kO) B(SF; + -C @ T: of behavior The BSpin g: @- ko)/~pin SF -&- J . M --% and (SF; M J @P P P P B(SF:kO) not do which of of splitting the into enter domains their parts those Proof. According to the J-theory diagram, these composites are maps - Theorem the Lemma is analyzed in following immediate consequences of 4.3, k fibres the previous corollary. the from of of diagram pullback induced It follows We and the J-theory diagram. 4.4, agree to write 5-I for any chosen homo- i that induce trivially monomorphisms, and therefore these composites (by finite- topy 5. a homotopy equivalence to inverse CP on homotopy groups. ness) isomorphisms, T: Corollary homotopic the composite to is kO) B(SF; + BO 2, At 4.9. Q9 Now following the yield J-theory diagram the of elementary chases in- 3 F BO B(SF:kO). --h BSpin 2 BO (u3)'1 Sullivan of Adams' work, which are terpretations based on ideas and results of QD I unpublished]. [72; > the k0) Corollary 4.10. At p B(SF: 2, composite TV: W + BO + is p_ @ composites following are 2, = p equivalences: At 7. 4. the Theorem : homotopic to both of diagram composites in the following the F BC~XBS~~~-B(SF;~O)XB(SF;~O) i B(SF;~O) "" ko) Spin (SF; (ii) "g- (SF; C X x (SF; ko) ---9- ko) 2

60 fit > the composite p 2, wL-+ - B(SF; k0) is vl: B0 4.1 Corollary 1. of Let basic fact in the theory following localization [23,V Ii 6, 7 or 48,~11]. the the both following diagram: to in homotopic composites of the localization Y . Q at H-space Y a connected denote of I L(ai(,,,r-l) v+-l 0 - BO --L+BO T -(SF; kO) v J- W map T, primes of set any For 5.1. Theorem the natural I V' vJ- I L Here are relevant routes to the splitting lower of the B(SF; while kO), of is an injection and is type. finite a bijection is f*Y if the upper bundle interpreted readily more are theoretically. routes with concerned shall generally be We the when Y, connected simply and J etc. The spaces = X C J J X = Define C spaces global P' P* of When T is the set p sense maps. unbased of may the in taken be I~rackets C fibre are often called Im J and Coker J. J is usually defined as the 2 = YT. The fact that (BF) 0 " * will allow us to ignore rational .~11 Y primes, 3 of BSO -+ BSO -1: $I 2; localized gives the same homotopy type as this at write oherence the localization of a classifying map t below. We shall f at f P p for structure. J2, but with a different H-space our In key role the of view played classi- its for and [email protected]) same letter *.tnd for an element of we shall use the study in the results above the groups of and the in JO(X), the present g by curly brackets we used earlier to distinguish stable from drop lying map; the definition preferable. is It is also preferable on categorical grounds, as was bundles or t~nstable bundles. oriented §3. noted VIII in explained be will and [unpublished] Tornehave by terms of In use We main technical result of as Corollazy 4.5 a substitute for the the image is in that anomalous fact we stems, choose to ignore the stable q in the 1.11 [2,3,4] analysis Adams' of and global version local mixed following 2. 1s instead preferring J), of image the to J (anomalous because q not in of that an F-trivial stable 0-bundle admits a reduc- the groups t,f JO(X). Note first of the periodic family not in the image regard q as the of J in element I Lemma lvn to Spin (compare 3.4). the (see Remarks (8jfl)-stems C spaces the defined first Sullivan 5.3). P' Actually, his (unpublished) is the fibre of f: F/O + [email protected] localized at C 2. Spin- a stable for equivalent are following The 5. 2. Theorem 2 Lemma 3.4. This definition is equivalent to ours by Theorem 4.7 and The over e l111ndle X. definition of new. BC2 given here is as trivial spherical fibration. is 5 a stable (1) KO(X) prf, in $Irq~ = such that ~spin[l/r](X) t 5 a unit There exists (11) 2 every integer for 2. r and E invariants 5. JSpin(X) and the 6 such that prime each For p, there exists a unit 5P (X) e KO FJ (iii) are bundles of to X section this In base spaces the next, and the be connected finite CW-complexes. theoretic To derive global bundle conse-

61 ~'1~'/1111//1/1 I// / 117 on defined was Recall that group $r Abelian the ibnd T: BO +B(SF; kO). p, there exists For an element q E KO (X) such that (iv) each prime 8 PP qr - is 1 difference The on orientations. action %(SF; k~)[l/r](~) by its E r(p). = r (X), KSpin in - $rq = P P P fr map classifying space the on ~$~/l B(SF; k~)[l/r]. the by represented . a map Bjo E r *, there exisfs as X -+ BSpin. If Proof. Regard 6 Theorem spherical 5.3. Let (E , t~) be a kO-oriented stable fibration SF/Spin BO -+ SF/Spin f: fE', = I Define 5. " qt' that such -+ X : ' 5 @' the statements. following over X and consider the J-theory diagram. Then (ii) holds by (iii) and (ii) trivially implies (i) The underlying stable spherical fibration is trivial. homotopic null is 9 Bjo holds, (iv) If Corollary (iii) implies (iv) by 4.5. For (ii)p p, there an exists (q element the prime (X) ) E %SF; k0) prime is null homotopic. each because its localization at P v~ P' of analog purely algebraic gave a local [15] the Atiyah and Tall (q p, such that (6 p, pp) = $r(q p, vp) - vp), r = r(p). +> (iv). equivalence (iii) extension the were approach in their points Key odd (ii) provided p and, each prime Statement (i) implies statement for P 3 3 operations by continuity of the Adams to p-adic y-rings p-adic on that 2. ($ - 1)fio2(x) C EspinZ(x), for the prime = Con- (+ l)i\ispinZ(x) - the fact that of use r and integers r(p) topological group generates the of all for primes p imply statement (i). (ii) statements versely, P algebraic considerations These > 2). p-Bdic integers p the of units (if 5 (e, p) is classified by T*( for ) some Then Proof. Assume (i). gained Sullivan's introduction of the geometric content with p-adic comple- 3 Im-r* under /1)* (B+ Irn C the stated 2, = At p BO -t 5:X map 8 - - of tions spaces 1. [72,73 he pointed out particular, the p-adic com- In that that note 3.8: Proposition and 4.9 Corollary proviso by the cer- proviso 2841 splitting algebraic obvious the of use by be split can BO of pletion [15, p. 2 C Corollary by I~(B$~/I)* Im(~v)* p >2, At = Z2) (X; H if holds tainly 0. 1 y- of the functor to p-adic rings obtained by p-adically completing i\io(x) and 1.10 3.8 4.3, Lemma by 1m(~++/1)* C lm(-rvL)* while Proposition not X p-adic to passage that me to seem does it for finite. Nevertheless, * since q(~+~/1) = 2.2. For the converse, Theorem and Proposition 4.4 completion would parts stable the in substantive additional information yield p for all implies (i) by Theorem 5.1. (ii) therefore and q = qe~$r P Adams' chapter. of work discussed in this and Sullivan's Let Proposition Then X. over Spin-bundle 5.4. stable be a 5 t cocycle by condition Bott emphasized also prefer not to use the We (6, and if spherical stable kO-oriented a as trivial is p.(g)) if fibration only , [4]. emphasized condition periodicity Adams or [22,72] by the and Sullivan each prime for pr(p)E, E 1 = p. KSpin (X) P P one only sense make conditions These conniders the when cannibalistic 4.11, 4.4 and 5.1, Theorems This is immediate Corollary from Proof. - pr of r(p). for general values rather than just for the r, classes These the of a chase diagram. and J-theory cocycle the conditions periodicity and classes give no new information, and a (special) unit in p be KO(X). Let p) Then (g , 5. 5. Proposition of formalizations as viewed be can new information. give no they why ' is if only and if fibration spherical stable kO-oriented a as trivial the trans- three results analyze The following kernels of the natural i primes all for p. pp Ir(p)p = by formations maps represented the kO) B(SF; q: kO), + BSpin gt BSF, + B(SF: P ."

62 immediate Theorems 4. 4 and 5.1, Corollaries is This from Proof. - 4.1 of the J-theory diagram. 4.9 and and 0, a chase analysis of makes rigorous The rest and in this section elaborates the following definition. This suggests the and am,plified by Sullivan $71 [4, Adams by proposed a speculative program that the theoretical framework envisioned by Adams It will emerge 61. [72, 5 be to J%(X) G = the SO, or 0, define Spin, For 5.9. Definition analogs leads to new bundle theoretic the 6 and E of d and e invariants of course, results the Of [5]. stems in stable of computations his in used diagram. J-theory visible in the are computations these Of groups course, the are abstractly isomorphic. JG(X) and Js(X) hon?otopy -ir*Jp can and the 3.5 Theorem from off read be Remarks 5.6. the In of Spin, the J-theory diagram yields a geometrically significant case .(P) BSpinp. the p-local exact sequence fibration J - For of BO P P choice, of isomorphism . the J:-rr.SO = -rr.Spin -, -rr.SF i of image > 2, read off the from can then be Define follows. E Q(SF;kO)(X) -+ : JSp%(X) as Definition 5.10. map The elements detects a*BOQp - r*SF e*: SF p. at splitting of SF-bundle p, a kO-orientable stable choose e , kO-orientation Given a 1 or 2 mod 8 that i > 0, p. E v.SF of order 2, p. where i m such and I1 cannibalistic localize p, and apply the at class c(+~(P)). The image of -rr.Spin. comes from an element of TT J which is not in the image of 2 i is independent of JSps(X) choice of p, the the p-component class in this of e* corresponds via adjunction to Adams' d-invariant (which Clearly a($)= ; ~(+~(~))(5,p). Equivalently, for 5:X -+ BSF suchthat ~nd assigns the induced homomorphism a map K-theory to real reduced of c *, choose r:X .+ B(SF; kO) that q'l; c 5. If such qr' r 6, Be. also sn+k an is the map e (which and Delooping infinite loop map) + sn). and thus, at p, prime each BO -+ 6:X some for 76 r x1/F then @ generalizing follows to arbitrary X, we can reinterpret this invariant as = r .(P). = kr/1)rp , C(#')S~/C(+~S~ - c(+~)(F~/ZJ (compare IV. 3.2 ). 1 + ) = ; c(+r(p))$ is a well-defined element "(9 . of (X) Therefore E JSpi (6 kO) For 5 Definition ) [email protected]@ E 5.7. w(5; = I(!$) define ~sF(x), E to for equivalently, ; g of kO-orientability to the obstruction the be We need one more definition. 5 = Be 0 5 e [x', BBO 1. E [x', BSF], 6(5 ) B the C(X) kO-oriented stable denote set of 5.11. Let Definition is defined on (a subgroup of) the kernel of d, Adams' e-invariant p) , cannibalistic local with X over classes (5 SF-bundles 6. Of course, the be defined on the of kernel and our E-invariant will for is , (5 if Equivalently, p. prime each p) (X) KSpin E 1 = (kkr(')) c P kO-orientable stable SF- Q(SF;kO)(X) of group latter kernel is just the 0 c(+~(@)) that required is : , a by n& ~:lassified F X - (BSpQp be it P defining Before we E , Theorems note that 5.1 and 5.2, together bundles. following with Lemmas 3.1 and 3.4, imply the result.

63 1, Ill I /'I ~ll//l/l 121 splittings of the B(SF; kO) It the form of the is immediate from that this a more of development conceptual section complete We withthe P classified is C(X) BC. by the space C(X) re- We 5.11. Definition by given than that escription of the functor uire some preliminaries. Theorem The 5.12. composite C kO)(X) JSpin(X) JSpin (X) QSF; Q9 onto monomorphically maps C(X) isomorphism, an is under E of kernel the ons 14 Define 5 bo, 1-connected, and bspin to be the o-connected, bso, orientation, of and therefore neglect bu and bsu to be define similarly, kO; and spectrum the of covers 2-connected = Q(sF;~o)(x) ~s~in(x)e c(x). case, each In kU. space zeroth the of 2-connected covers and the 0-connected clause and first holds by comparison of Corollary 5.8 Proof. The are Is the one suggested by the notation. (Warning: bo and bu usually taken - Definition 5.9 with 4.4. Theorem of equivalences the clause second The of a useful notation. ) ns kO and kU , this being a pointless our waste [X, [X, by a monomorphism is BSF] the splitting -t BC] (qn)*: since holds away all spectra localize 2 and 2 r. r Fix 5.15. Lemma from 4.10 and ] B(SF; $0) Corollaries 4.9 and BC (which show that n*[X, of P P k0 + k0 @'-I: bspin. k0 $ir-1: uniquely to -+ lifts Then intersects onto the (Bodp] and trivially) since C(X) clearly maps T*[X, lifts uniquely Recall bo. into a map to $ir-1 obviously Proof. E . kernel of 0 if non- the is unique R S 7: Bott that periodicity implies that RmS -t ' m Theorems particularly illuminating, of be Comparison may 5.3 and 5.12 trivial map, then at the prime 2. i 1 s SKOAR 1 KOAS = z KO lhq:z~o KOAQ~SO -+ We discuss the relationship between Adams' e-invariant and our m of fibre the equivalent to is -invariant the denote KU and KO (where KU + KO c: E in the following remarks. $2) (LI connective spectra associated to Passage spectra). periodic Bott ; Remarks 5.13. Let straightforward :SF E X = E Jap. JeP= -c A t P P in groups homotopy behavior on same the with map 5:;SkO yields a kO the the chase J-theory diagram allows us to identify of on E-invariant 1 as degrees non-negative ;i uniquely to a map lifts obviously 1 ~7, and with Ker(Be)* C n*BSF homomorphism induced the kO T onto clearly maps adjoint 7: k0 - Sl bo Its bo. -c ZkO ?: n e* 0 * n*JQ2 n*B02) )+ E*:Ker(n*SF4 Ker(n*J*-- -> n*BO * @ and ZZ re- with for cohomology in coefficients Write Slbo H . n ZZ = 0 1 2 the elements p. to avoid taken being (the kernels and their of Remarks 5. 6 , H*k0 = denotes A where Steenrod ASq t A/AS~ the mqd 2 call that 0 ). an hand, other the On inspection J n of the images in 2-component of = Z2 and H1kO = In 0 particular, HZlcO. H kO = (e. algebra [7, by g., 3361). p. *@ will e-invariant [5,§ 7 and $91 (denoted convince the reader that Adams' real bo gives an exact sequence bso fibration The 'k(Z -). ,I) - 2 Indeed, it can be ek eR in or [5]) admits precisely the same description. 0 bo] + bo] [kO, - bso] [kO, .+ - kO [kO,R 0 H seen in the calculation that Adams' retrospect of e-invariant the image of on fibration The surjective the which first map is in by the properties of 7. the [5,§ 101 amounts to a direct geometric comparison between two in invariants. J

64 123 bspin gives an isomorphism (2 'k - bso - ,2) bso] and [k~, bspin] [kO, - are B(SF;j are of fmzte, there ) no i,rnl problems and ~roups obvzously 2 P the conclusion follows. composxte we conclude that the Definition 5.16. Define j p r HJ to be the completion the of at =) c( B(SF; -..-EL jp) B(SF: ko) L~s~in~ P fibre of . kO for use bspin and define in VIII, - j = X j +'(~)-1: Also, Consider the BC - ) j B(SF: . I: lift a results There homotopic. null is pP 3 P P kO be tk - bso and to -1: iji of fibre the j02 of 2 at completion define RB(SF;j ): = ) j (SF: following which in diagram, P P define BO +3-1: of fibre be the to J02 - 2. at BSO The use of completions is innocuous hem (since groups homotopy the spectrum. rzng a j is to in finite are serves ensure that and positive degrees) Indeed, 2 that j we shall prove in VIII. 3. is the a ring spectrum such that P - kO (completed at p) is a map of ring spectra. K : j map natural P 5.17. The spaces BC and Theorem j) B(SF; are equivalent. We e*: T*SF - "*JBp is that a split epimorphism by have Corollary 4.6. Therefore monomorphically onto T*C (ncfZ~)* mape and 4.7 Theorems irv 4.8, %(SF; = C(X) j)(~) P "*(SF; j ) monomorphically onto maps In the bottom row, iicr c*. P fibrations. spherical j-oriented stable of the group is and an isomorphism. monomorphism a is (ni)* Therefore thus e*. Ksr j By w. the zeroth space of 11 is equivalent to J X . 52, P (P) ' P is an isomorphism. conclude that <*:"*?(SF; j we - "*BC inoiooping, ) P P of as 1 The VIII 53. by H-space, an the J is 1-component its -component @P ' J is j of zeroth space J X = a give the projections and homotopy com- @P @ bundle the or^ 11. bulltvan's analysis of topolog~cal mutative diagram 7 9 J BSF > - B(SF;j) due is theorem baszc followzng The to Sulhvan. CD 3 STop. Theorem 6.1. There exists The a of g kO[l/Z]-orientation - of 2 from away Ic~a;&lixation associated to BOB[1/2] - :F/Tap f the H-map an - ii llSTop kO[f/Z]) is equivalence. B(SF; Actually, g that proof the is proven in first The statement [72, 561. B(SF; j) is equivalent to X B(SF; j ) We that conclude ; P P P that see to easy is multiplicative there.' omitted is H-map nn irr is g It spaces at p. A j -oriented Lemma 2.4. Fix r = r(p) and complete all P an 7 of the discussion as saffices this torsion, however, and tnodulo for in fibration regarded via when C(X) clearly lies stable spherical the homotopy j - kO r: a kO-oriented stable spherical fibration. Since as P given I. proof will be A in Theorem I. 16 below.


66 127 126 following equivalences. p > 2, the At are composites 6. Theorem 8. Proposition 6.4. homo- is diagram following the r, from 2 and Away - -1 i commutative: topy BC (i) BTop. (BTO~)~ . 7rXBiXq g XBOXF/TO~ XWXW'-BC P P 0 B - B kO) (SF; BTop 8 ( C x nw Xmyl~nvi Stop. *C XSOXQ(F/Top)~17r)xixn~ (S~op)~ 11) P - f - C - TO~O x TO~/O L TO~/O. x N (iii) '~(g-'")~% BO Bi . - P P . BTop L B(sF;ko) The odd primary parts of Brumfiel's calculations [24] of T*BTo~ and 2j-1 . following 1s a unit in Z if 2j z 0 mod (p-l), the Since 1 - 2 can the diagram. and theorem the from off read be T:~TO~/O (P) local theorems corollaries, and r(p), from = exactly result which r in At p > 2, the composite qv : W -+ F/Top is BTop -, 6.9. Corollary the and same,calculations their diagram chases that were used to prove 1tt)rnotopic to of the composites in the following diagram: both 4. analogs in section composites following equivalences: > 2, the are At p Theorem 6.5. 8 w A' w W~BO-BO -W , ~-LBO-F/T~~-W, '€3 I I A BO- BOX BO xrx (+r-')-. F/Top X BO O w x w- p At 6.10. Corollary W1+ BTop the 2, > composite is vl: Bin + BO vxvL wxwl-~ox~o of Itornotopic to both diagram: the composites in the following and c3 6. diagram is a pullback Corollary in the following the 2, > 6. p At category: homotopy - above the results theoretic interpretations The bundle evident of are from bundle all Consider diagram Iic I the previous section. of the arguments and d and p N~") at and define of to the localization be Define N every that 2 asserts 6. Corollary sight away 2. tl~cories in from as localized P P P the ;ijr(') of and xr(') be to localizations p. at Top-bundle (5, y(g)) for some has the form I* O[i/2]-oriented stable F-bundle 5 and and that two stable Top-bundles e and i ' are equal if (g , y(g)) Corollary equivalence: 6.7. is' composite following the 2, > p an At (t. are equal as kO[@.]-oriented stable F-bundles. (Here, away from ',ti(?)) .', we ) F and Top but think in terms of the integrally oriented case. may write

67 R(SF; monoid on a stable F-bundle. to Top) kO)(X) may be interpreted as The Adams acts on MTO~(X) via its action on the Sullivan operation +r and the JTop(X), E JTop(X). defined on as regarded thus be -invariant may +r on of action the More precisely, r). 2 and from (away orientation N N all set stable Top-bundles over X of of whose the as may be interpreted C(X) is z*. to equivalence the along KTop(X) to be transported kO)(X) K(SF; [ 4- local cannibalistic c(+ classes then may 12 5. Theorem are trivial rp)p 5- group BOd in action of Similarly, the on the +r of (special) units [X , KO(X) E be 1 follows. as interpreted acting +r trivially Then, with to [X+,F/TO~] may f *. transported be along 5 on the &(x), transformations induced by BE, maps Bj: BTop - the 6.14. Away Theorem (X) JO JTop(X) C JO(X) composite the 2, from i 8 i- commute with the +r (by Propositions Bi: -C BTop, and q: F/Top -F BTop BO monomorphically 11 an isomorphism, C(X) maps onto the kernel of under E [ The following analyze three results in 2.3 cases). two last 6.4, 2.2, and the homotopy equivalence, and passage to fibre therefore C of kernels the these transformations. Again, as proofs are the same for the 5. of results analogous the Section a precise Remark 6.15. analog For what it is worth, we note that there is the Theorem Away from 2, 6.11. following are equivalent a stable for of is slightly plays B(TO~/O) which in 6.14 Theorem to The BE. role the proof i Top-bundle c X. over : - into taken be must more complicated, but account, because rational coherence fibration. as trivial is 5 spherical a stable (i) i I that, again is conclusion the of image the 2, from away prime For each odd p, there exists an element q (X) E ETO~ such (ii) P P - = +rqp that qp * 3.. = PI. P of a direct is summand of the image Proposition over be ( Let 6.12. a stable Then, 0-bundle from away X. 2, is and only if trivial = 1 E KO (X) for as a stable Top-bundle 5 if er(~)e P - P odd prime p. each in This remark also has an analog the with complementary summand C(X). J-theory the next section imply that J-theory case since the results of the 6. Let Proposition 13. , (5 Top-bundle F-trivialized an be T) stable corner. right-hand a lower admits diagram over from Then, away X. Top-bundle if and only 6 2, is as a stable trivial odd +r(p)(c T) = (5 ,-I-) for each for prime p (or, equivalently, Jlr(p)~ = 5 , if former the Remark 6.16. We have used Top and F instead of PL and G since P P 5 is that unit of KO[@](X) suchthat the Sullivan odd where prime each p, from naturally our general context. Stably and away into 2, there is - theories fit T e is from of S and the orientation induced by the cup product of orientation of no Unstably, Sullivanls is the limit of ; distinction. kO[~~-orientations the canonical orientation of the trivial stable F-bundle). ; SPL(n)-block bundles g(n): BSPL(n) B(sG(~); k0[@]), where the classifying r use of by or [47] of methods the by either constructed ! space on the right can be the 6-invariant to interpreted be may as 2, Away from obstruction the G/PL Brown's theorem. Haefliger and Wall's result [32] that G(n)/PL(n) - i a topological structure (that is, a reduction of the structural the existence of ! r e i-

68 The proof of Madsen, Snaith, and Tornehave (MST henceforward) an unstable comparison diagram is for n 2 3, together with an equivalence real following = p at case 2 the in complement except yields the well equally Lemmas and 2.6 of use and arguments sequence Barratt-Puppe obtained by 2.7, result the where [38]. Ligaard to due is Note, however, that is an equivalence for every nk 3. show that x(n)[1/2] one used most in this remark is not the of version bundle block the PL(n) T-local be connective E T-complete or and Let D 7. 2. Theorem work geometric piecewise-linear topology. in relevant to !rpectra spaces of which have completions the zeroth each p c T equivalent, at Then U of those respectively, to BSO. and Spin or BO and SO or BU and 0. = E] I), 1 diagrams Infinite loop analysis of the main 5 7. h A' various determine of to In the which splittings we have ob- order of proceed analysis an to H-maps or BU -C f: BU which MST then P P A Such uniquely be a map can loop tained we must determine are actually splittings of infinite spaces, I: BSO -C BSO maps. loop infinite fact in are P P turns It that out maps. loop infinite are our main diagrams maps in which = are H-maps and fl is written f 12 f the + in f *', where f 1 and f2 form homotopy theoretic arguments, which can be thought of ultimately based as basic observation ljrlme that f is to +p (in a suitable sense), and their is how on the of BO together ties periodicity Bott tightly p-local k-invariants = if .,!I a essentially then is f (since 0 infinite f2 only and if map loop assertion This the has with coupled BU, and yield a great calculations, representation theoretical of i/~~ Itnear combination the with r prime to p). consequence. I~rllowing deal of information about to due are arguments relevant The question. this To begin Madsen, here. will be outlined and [42] Tornehave and Snaith, T-local X, Y, and Z be T-complete infinite or Let 7.3. Theorem have proven the following the Adams- of maps for analog these authors with, of at spaces I~>IJ~ each p BSO. 6 T are equivalent to those whose completions Priddy unique deloopability of spaces result, Theorem 4.2. Their proof is frX -c Y and gr Y -. Z be H-maps such that gf is an infinite loop map I.r:L A A n = ] BU n), [K(Z that fact the on for based 0 2 3. An alternative proof g a5 nltd either f or equivalence. is both rational infinite loop map and a P P' possible. any [8] based Priddy and Adams of techniques the Let T be on is infinite loop map. g an f is map or remaining I'hen the set primes. of of the results of this section In all 2.13 our Theorems , 11. By 2.13,II. suffices to and Theorem 4.2, it 2.14, show II. Proof. - A and follows localizations for result the that show immediately 2.14 11. at T are infinite For spaces. loop as BSO definiteness, Z and X,Y, when Illis P p from the result for completions at p for e T. f 1c.L and Then equivalence. map infinite loop an be a rational f 1' = f or E be T-local T-complete connective Theorem and 7.1. D Let implies 0 = fg2 But fg S fg 12 I! = gl + g2~p, and fg = by = fgl. g2 = 0 zeroth completions the spectra at which have spaces of each PET to equivalent I .r.lnma 2.8. fi fi [D,E]-[D E is those of ] either BU or BSO. Then the natural homomorphism an criterion above for determining whether The H-map BSO -r BSO 0' 0 P P a monomorphism. In infinite loop map will be interpreted representation theoretically at an by results of representation pair following prove the MST 1. 5 VIII end llr: t of

69 r 2. r [3] for pr; the result for 8 r p (+ 0 2f +4) theoretical on calculations based as the non-trivial delooping. Define J @P the an infinite loop space to be follows. g J -t SF -, C shall construct infinite loop We an fibration fibre +'/I. of 7.4. Lemma pr: BSO&/~] is an infinite loop map. BSO[~/~] - @P P f By the following VIII p> 2 in VIII $4. and will show that 53 In it splits when Lemma 7. infinite loop map. an is l/r] [1/2, BO -r BO[1/2,l/r] er: 5. 60 basic result Snaith [33;70, $91, this shows that, Hodgkin to and the eyes of of is Theorem 4. 2. following only requiring simpler, The analog J to equivalent is SF K-theory, [ 49P ' c. infinite loop map. is +I: an [l/r] -r BSO [l/r] BSO 7.6. Lemma N * N* F- '8 '8 and K (C ) = 0 KO (C ) = 0; there are no non-trivial 7.8. Theorem E. P 'g. P an gives 4.2 Theorem equivalence prime to p At Proof. r, C - BSO -r on either the space or the spectrum level. C maps e- P By the 3.7, Remarks in argument 5: spaces. infinite loop of BSO + BSO % now of analog following prove the can We which and 7.2, 7.1 Theorems r -1 +' is homotopic to the . map &I 5 infinite loop L le due to Ligaard at p > 2 and to at somewhat are proofs Their 2. = p MST t at the to convenient is it point, this At insert a remark relevant only i difficult, but give more precise information. more 1 prime 2. Let be or T-complete infinite loop space a T-local X 7.9. Theorem 1 N* I- Hodgkin 0 = (~(u,n)) KO that [12] from and Anderson Recall Remark 7.7. an Then p T is to that of E equivalent each at completion whose BO a i By for 2 2 and all finite Abelian groups n. n use of II. 3.2,II. and the 2.10, i= most spectra. map one H-map f: J - X or g:SF -* X is of zeroth map of at the non- s plittable fibrations I- % of n. X replaced with p, at by work may in We Proof. BSO, view 2.13, 1- b0 K(z2. 1) and bSph -t bS0 -C X(z2, 2) , bS0 + the 11.2.14. follow g will result for Clearly 4.2. Theorem and Lemma 3.1, easy deduce that it is t~ ! With r following the consider r(p), = result for the from immediately f. [X(Z~, 0, n), bo] = 0 for n 2 0 , 0 n), k~] = for [1((z2, 2 n diagram, the are infinite loop fibrations: which rows of n), 2, 2 n for 0 = [X(Z2, bspin] bso] = 0 for n li . [k(z2,n), J-1 BSO - - BSpin -----3 spin Z2 Z and [X(~~,l),bspin] = [X(ZZ.O).bso] = 2 ' P last spectra maps both cases, two of the In induce the trivial map and BSO - 0) K(z~, on zeroth spaces. BSpin - 1) K(Z2, [email protected] In we shall prove that c($~): B(SF; kO) + 93, is an infinite VEI loop map at p, where r = r(p). Thus the fibre BC of and its loop c(+') P space are infinite loop spaces. C Define clir/l = c(+') T: BO%+ BSpin P '8 is the one coming On BSOw this delooping map at as an infinite loop p. from equivalences coming loop infinite are 5 p, a, 4.2, Theorem The maps of 7.6 in from Lemma view 7.1. Theorem definition = 2, this fixes When p

70 and the T loop maps, and 5' and cut are infinite loop the are natural infinite sequence from (e. g. Barratt-Puppe maps coming arguments [48, I]). Clearly F,O yp BSO an is 5' equivalence. yields [33,4.7], the top fibration By Hodgkin and Snaith sequence an exact infinite loop the is trivial BSO + 4.2 and Theorem 7.8, prq~ : C By Theorem PKo(Bso) o -. o - PKo(Bspin) a ) 2 PKO(: P C9 P = as prq such that qL an BSO -+ B q: map hence there-is map, infinite loop of primitive elements in p-complete K-theory. but KU, the (They deal with @ P rr Remarks 3.7, infinite loop maps. By By ($r/l)pr. cr " p . Thus q(Lpyp) r;r for result KO follows. ) J f: Let BSO infinite loop map + which is be an @P Clearly is infinite loop map. an 7.3 and 5 7.4 and 7.6, Theorem y Lemmas as a map of spaces. trivial trivial is as an infinite loop show that f We must P P -1 y a homotopy and fy = pr(< equivalence, ) 5 y while f and is an as % = f 7.2, infinite loop Theorem By map for some infinite loop map. PP PP P SPyP 1 both the from restrict trivial the splitting map on C It follows to P 5 )- y r(b -+ BSO fibrations and are 0 of negatives = [spin, bso] (because BO 5 map P 11 P' 69 E/O 4.8 f is homotopic to the infinite loop map in that Theorems 4.7 and of [48,X1]). the stable category in cofibrations primitive the exact sequence By of H-map is there 2.10, an Lemma 2, = p when H-maps) e., (i. elements and, As M such that y$e ($r-l) =Tag. f: BSO -t BSO By infinite an is f 7.3, Theorem following is The of diagram a commutative (globally) Theorem 7.11. 8 map. loop fcr't' = %@5(41~- 1)lr is the trivial By 7.1, we Theorem conclude that maps: loop infinite and spaces 9 infinite loop map. Since by a monomorphism is bso] [yQ2, bso] 2, [b (el)*: - -2 - BSpin Bj + sF/spin BSF SF infinite loop map. trivial the also 7.7, Remark is f These results allow infinite loop analysis of the comparison diagram parts and the left 7.9 Theorem By the previous proposition, square is Proof. - theory. bundle topological for analog its of and of diagram J-theory the The in out pointed As infinite loop a and maps. of diagram commutative spaces was noted following result by Madsen, Snaith, and Tornehave [42]. a Barratt-Puppe an gives spectrum level the on argument sequence 2, section Proposition 7.10. f: infinite loop an (globally) is map. -c sF/~pin BSOQ infinite loop map com- squares two the right makes which kO) B(SF; BSpin- g': Proof. suffices with result the prove to all II.2.24, and 11.2.13 By spaces it On some for rh = g' - g space level, the level. space the infinite loop on mute at p. completed it suffices to consider f: By Lemmas 3.1 and 3.4, F/O + BSO @ middle square implies that h: -, BOW However, commutation of the BSpin (even the at compo- p of HA in cofibre the of space zeroth the B be Let 2). = Lemmas by 0 h Therefore = h* 0. g* that thus and homology rational on (gl)* = site map infinite loop C L: F/O and let 5 : F/O + BO be + (SF; kO) .+ SF - P P Thus g g'. = 4.2. Theorem and 3. and 2.8,2.10, i map. natural the Theorems By and 4.7,4.8, to an equivalent B is 4.2, BSO as an is of equivalence Corollary 7.12. At p = 2, the following composite space. infinite loop the J-theory diagram Consider the following part of infinite loop spaces: with r r(p) = BC2 X BSpin %B(SF; k0) X B(SF; kO) AB(SF; kO).

71 ',is,') _I I! I ! 137 the II. work on may we 2.14, and is an the equivalence of p> 7.13. At Corollary following composite 2, Proof. Again, by 11.2.13 - With :F/O as in the proof of Z, B ' infinite spaces loop : 2. p> level, ()-complete P I. ixvxvL kO). AB(SF; gXT>~(~~;k~)3 "' BSpinX BC BO X BCpX - W X W the following in main diagram oposition 7.10, consider the part of P c3 cannibalistic the related J-theory diagram the universal of parts Those to r 6, oction = r(p): class be analyzed on the infinite loop level in VIII 5 3. ~(4~) will of parts J-theory diagram depend on the All remaining Adams the has 1411 Madsen shown on and thus conjecture yr: BO + sF/Spin. so or 2, at localization of that y3 its that chosen cannot be that B 0 B (See also seems [26,11.12.2]). Nevertheless,it is even a3, H-map. an can r, yr be infinite loop an as chosen 2 and from away likely that, following conjecture The is even a bit stronger map. 1 0 B The complex Adams conjecture holds on the 7.14. Conjecture €3 r, the composite for each That is, level. loop infinite space the trivial F/~op is C Bio~ : 4. Theorem 13y 2 and 7.8, Theorem P BsF Bu -LL BB : hence B ' F/TO~ map, loop there an Infinite is infinite loop map f map is as an infinite loop trivial when localized akay from r. Z, = Bi as infinite loop maps. Now 6 that auch (Spyp) 5 -- h P The proof it 11.2.13, suffices to work one prime at a time. By y 5 map, since infinite loop an the by map infinite loop is an is P P - as an infinite loop space each at VIII in as that SF splits $4 X J C map infinite loop an by is thus f and 7.10, Proposition proof of P P not is it but SF, -+ J prime p will give explicit splitting odd maps 7.3. and 7.5 Theorem Lemma P maps these of) or whether known homotopic to (some choices are not Away from 2, the following a commutative is Theorem 7.16. diagram. in a the J-theory of infinite loop spaces and maps: diagram P we have the Turning to the analysis of BTop away from 2, also was following analog of Proposition 7.10, which noted by Madsen, and Tornehave Snaith, [42]. i - ? 9 ----r BSF SF BOB -& B(SF; kO) F/~op[l/2] : f infinite . 7.15 Proposition -c BO is [l/2] an @ I map. loop 1- spaces. of equivalences infinite loop are 7 Therefore 7 and i. Remarks following See the VIIL discussion 4. 6. 1. i i

72 Proof. square commutes on the infinite loop Again, the left - ring spaces and bipermutative categories VI. E 03 give sequence Barratt-Puppe hence 7.9, Theorem by level arguments - a map kO) squares com- ': the BSTop ' B(SF; right which makes two g level. space the infinite loop on mute for 2 2- rh = some map E space is, essentially, an H-space which is commutative, An a3 of h is null homotopic. h: BSTop " [email protected] , and the rationalization It An coherence possible higher all homotopies. up unital and .i~sociative, to follows the by [BC the fact that BSTop, of splitting of use Bod 0, = I.: respect to with products, space two ring E an essentially, is, space P' a3 I I) g'. Note that, despite and -- that h Lemma 2.8 and 0 g r. the that ~~III. additive and the other multiplicative, such the distributive laws are played by BSTop on product the role its in splitting, this argument defini- The precise up to all possible higher coherence homotopies. .x.tl.isfied an K-map (again, because [BC does not depend on h being BOd 0) = Some of ele- consequences definition, and the ton will be given in section i. I P' to unpublished pr.ove Sullivan's suffices than more therefore and asser- will 2. section in given be examples, 881vntary tion that which has (a H-map an is g nowhere been used in result a category is category monoidal symmetric A is which a product with our work above). It associative, to up unital and .*~~irnutative, < natural isomorphism. coherent .It.f.a equivalent which d classifying space the category, permutative an s rmine as splits at BSTop 7.16, p Theorem and 7.13 Corollary By symmetric monoidal 11, ,tn En, space. A symmetric bimonoidal category is a fact, together with X W X the WL as an infinite loop space. This BC P multiplicative, other . .trcgory with respect to two products, one additive and the 53,has grasp on BC as an firm infinite loop space given by VIII been P natural that coherent to up satisfied are the distributive laws isomorphism. -s~c.Ii classes to obtain precise information on the characteristic used of rletermines an equivalent bipermutative category, the classifying space 11 topological in stable for 2) > (at p [26,II]. bundles definitions, and proofs, given The precise will be ..:iiich E an is space. ring to ~octions 3 and 4, and many examples of bipermutative categories will be 8,) in 5. altt~l~layed section be will ring spectra E ring spaces and E between The relationship 0 cn The in chapter VII and applications will be given in chapter VIE. ~i,.~t.rrnined ring spaces has been studied in [26,II]. ~~~~tttology of E a3 ring spaces \ I. The definition of Em As will be made precise below, an operad ti a collection of suit- is ,t!bly interrelated spaces c(j) with actions by the symmetric group X.. J

73 IlI,I I 1:' ~, 141 (C, a monad determines p, 7) explainkd As operad an [45, in $21, an E operad if the I: actions are is free and the spaces g (j) are con- (= j a, -space is equivalent to that of a ,C-algebra the that notion of a such 3 K(X are of so that the tractible, &(j)/zj 1)'s. An action FJ orbit spaces j' on an opelad b 1 action operad an of an of We shall define a notion . 3 in based spaces) is a morph- X 3 (our ground in category of & on a space in to restricts a monad the category C in such a way that Thus & 3 1. [ ism of operads EX, function is t&j) F(x',x) the + space of where 5 bo-space inherit will CX the space X, -space a structure of a for based maps and evident the in structure operad [45,1.2]. an given is way 0 q:X morphisms will auch that the unit CX -) CX and product p: CCX -c be E An (X, by action given X 9) 8 a space with a some together is space w E of an over be to t, algebra bO-spaces. We shall then define a ( h)-space by c e &(2), product given 8 (c):x2 ,X for any Ew operad 4. With 2 The requirement that the additive action the monad C in b [ gel. is which is commutative, associative, and unital up to H-space an indeed X g ko-spaces encode succinctly will the distri- X be a of -) CX FJr morphism higher coherence homotopies all possible p. 41. [45, a ring space will be ( c, fi )-space where 6 and h. laws. butive An E E An ring space will be an Em by two actions respect to with space m m 0 k are E operads. operads. interrelated E These actions will satisfy the distributive laws up m m &-spaces to )-spaces (15, kl of think from is It useful to the passage these possible higher coherence all to although implied homo- pies, homoto as resulting from of ground from category a change to 3 .b [gel. topies not be made explicit. fortunately need be 8 . Actions by will assume given operads and Thus two definitions the useful of The details in and rigor for necessary are The 8. corresponding additive and will of be denoted by thought as always in and homotopy operations [26] (work operations homology of the study basepoint and smash be be will denoted by zero and wedges products will out- For our present theoretical purposes, is it the conceptual progress). basepoint. this taken with respect to by Actions thought of as will be h recall first We the definitions operads and of that is line above crucial. multiplicative will always be denoted by and base- corresponding The . 6 operads, by actions 8 I be will point denoted by one. As in $1, IV is it - .& that to insist convenient ue Zk, For j 1 <_ r < k, and let 2 = j1 t . . . t jk. 0, Let j Notations 1.1. zeros. spaces 3 that Recall have of denotes the category together X spaces of be letters that to permutation which permutes j . u(jl,. define I: E jk) , j er SO X that and with cofibrations 9 of the category denotes [gel by letters of the k blocks determined the given partition of j as r permutes non-triviality, that, for to observe important is It e). (X, /do-spaces I:. . . . rlb define of T~ to be the image CB E T E I: k letters. For r jr J ' X. in and one must lie zero different components of in the Indeed, contrary I: in T , . . . . X . I:. of . X.X. inclusion evident the under ) (T*, k J, ' j 1 and 6 x) (g)(O, e2(g)(l, x) by zero and one would case, translations the left 2 a collection is operad spaces An 1.2. Definition k(j) for of Since and, trivial is the first map homotopic be (for any fixed g (2)). e )5 , * maps with together j with 0, 2 point a single E(0) case in practice), the second map h(1) is is connected (as is always the if the to homotopic identity it be [45, p.41, follows that X would contractible.

74 ~(((O('((U<:S(ji,...jk) ... j u{ii,. . . . ik) = {i u-i(i)*-..ni 1- u < (k) u- . . , dk); ei,. . . , e Y(Y(C; di,. j . +jk) = Y (c' f +. . . . fk) . . , i the given isomorphisms of S(ji,. . . , jk) and ~(j Via ) *j *... u- (k) f = (dr; e where y if (or * j =O). Sjr . . ) J jiS. ..~j~-~i-i'*''*~. S. Z. of element an as u < ji, . , . , jk> may . regarded be j}, . . 2,. {i, with , i J @. E 2. , define T~ T . . @T~ E Zj For . by k If c E (k), then y(c ; i ) = c ; (b) d E C(j), then y(i; d) = d. Jr if (T @~~){i~,...,i 1 = {-riiir...,~ i 1. iQ9... k k k and dr E & (jr), u E Zk , E T 6 X , then If &(k), c (c) jr di2.. . , ~(cu; = y(c; d u-i(i) ,...,du-i(k))dii,.. .,jk) dk) i hri Given non-negative integers k, jr for and < r < k , i. 5. Notations and permuta- that to be .} h , j v{k, = v define , jr 5 i 2 i and k 5 tor i ( r . T~) . . . 6) )(T d , . . . di, y(c; = dk-rk) , . . di-ri,. ~(c; 6) r rl k i of t the ion set of con- 8 an operad jr a space X of k on action An 1.3. Definition ) hri 6 ( TT ( = hri) .C X . 8 maps -equivariant Z sists of + (j) such that eO(*) is the X xJ X r=i r r=i i=i ' j j . . , jk) S(ji,. IE * e X, 8 (i;x) = x, and, if c basepoint c(k), dr e E (j,) i < r ( k, for i the which corresponds to letters of two ordered sets (where the comparison and x for i(s(j +... +jk= j, EX i denotes the ordered disjoint union) fi di,. .. ,dk);xi,. . . gj(y(c; = Bk(c; yi.. . .. yk) . ,x.) J jt jk where ,..., \J and S( hii. ..., \) S(hii i= i i= i i .jk) . . . I~s(j,. (or * if jr = 0). (dr;x = yr 8. ) S... +jr jiS...+jr~i+i""'Xj i r J th I the of with . . . , ak) i, i < a, 5 hri an obtained by (a element sending r We the multiplicative analog i. i, and distri- require Notations of b ) of the second set {b summand of the first set to that element k 1'"" permuations, to define actions of operads on operads and of operad butivity . +a . hri + .. br that = 5uch hri + pairs on spaces. r-i k A of an operad on an operad & con- action An Definition 1.6. jk) i, j 2 sequences let S(ji, . . . , all denote the set of For Notations 1.4. slsts of maps I = {ii,. . . , ikJ such that i < i 5 jr , and order S(ji,. . . , jk) lexicographi- A: X C(ji) X ... X C(jk) + c(ji*-*jk) B(k) [or formulas: kk 0 and j ) 0 subject to the following dl

75 I I 1 111 145 assume Henceforward, ). given an operad pair (c , E &(is) for gr If g E fi (jr) for i l r i k, and cs E )d (k), (a) . i then , jk + 5 . . ji+ 5 s Let 1.8. define E 10 j and 0 2 k For [je]. 8 (X,c) Definition I . ' . . gi, X(y(g; cis..., c gk): where @ ) 0). = j (or*if . jr X(gr;c dr= j +...+jr-l+l''"'C . formula the E by &(k), cr e C(jr), and +jr . . jl+. 1 y E X for g following E (hri) E h (k), cr e & for (jr) g dri and k, ( r 5 i for If (a') iiiI.jrr then k E if y where, ). x , . . (xrI,. = . . , (xli = . xK5) E X . . then yI i rjr k< ?k in C monad the of construction Recall the 3 from [45,2.4]. where dI=?,(5; d,;,, . . .> dkiJ [gel, the maps gk an action induce For (X,{) E .Q Proposition i. 9. = er and ,\) . . (diii,. dri.. d . y(cr: = . . 1. d I rjr of morphisms are CX + .Q:X and CX + p:CCX that such CX on h of b, then If CE A(l;c)=c. istheunitof k(i) i~ c(j).and (b) C monad the the Therefore bo-spaces. in monad to a restricts 2 in k )= istheunitof , then X(g;i If i. g~ (b') h(k) ad i E e(i) ,I. /.. [ 3 category then el), and u , E Zk . (k), h E g E cr If (c) (a') of ahd (c') By Proof. 2.3]), u. the degeneracies (applied to [45, i,...,jk>. )u

76 0 one), g t h(j) and Ei = for or and ordinary the is product (where the i ($ , A commute. of map 1 of a map both is which a map is )-spaces 0 any for ~o-spaces of a morphism is X -. c:S a For X. -space k ~o-spaces. of -spaces and a map 0 e let X, &)-space (G, map the induced denote also notions of the Since 1 and of C-algebra in -space are equivalent 2.81, [45, it follows immediately that the notions of (cso,ll>5) (x,e, 5) - (C , ?j )-space and of C-algebra in are equivalent. 01 h [a the called is e )-spaces; h (6, of X. of unit Definition is An 1.11.. Em ring space a unit is analogous e to the Of course, of R + Z R. a ring e, ,k$ )-space where ( (k, operad pair. &) is some E is the If G denotes the monad associated to N , then GX free m operads, When 6 and b are E fJ -space generated by X. a space 0 a morphism We have not defined and do not between need any of notion CX [45,3.10 GX 46, - of 6 (C and X G)(X) the homology isomorphisms different over spaces ring Em E operad pairs. m be used interchangeably and can 21 thus both A. that show C and G can The on !$ )-space gives , ./3 -action rise a (c I,e thought additive. as of examples Units; 2. pairs operad of lo exponential units. curious and useful here We display the definitions and of a few consequences out point a 2. any given For -space. kj element be 5) (X, Let 2. Definition on another; these should operad help to by actions of examples some one r SO chosen any to 1 and 1 to 0 by sending X - : e a map define %X, s motivate our general theory. a based as regarded SO With r. the component base- point in with space & -space generated by X. For free is For a space X, CX the map the denote also induced e let and GSO form 0, point same the categorical reason a [45, 2.91, if X is -space, then CX is (GsO, 5) (X, - ll) BO by X. In other words, /$)-space ( k, iJ)f gel generated , (c free the if of unit exponential homotopy Fd ,,I called is X. e -spaces; to Up an denotes the category of function the then )-spaces, fi , (c is independent the choice of the in the com- point of e b -maps, LI~rough r. ponent Hom (y, 5)) Horn + ((X,5), ((cx, CL, 5), (Y, er 5)) (c* )[Lfel h h [ gel the the as integers non-negative of 2' monoid of Think set free composite the to - f: X sending by obtained Y , )- maps e The 1. element the by generated (under addition) for a (C;' e 'Y CY . Cf. CX f. Into the underlying from of maps analogous to the are Hpace monoids Z Y. g: CX for -, q 0 with is a natural isomorphism g inverse g -c ~nultiplicative R. a ring of monoid 0 several are There examples algebraic of operad pairs, for which Definition 2.1. N. ,5) is 5, 1. (S a As observed in b0-space with Let an de- fi and h(j) are discrete. c(j) tile component spaces and Ei' E.) E. .. ., = E,. -.. Ej(g)( J J

77 monoid note of [45,3.1] such that an 4r) -space is a topological operads the and an n-space is a commutative topological monoid. Recall that group with Grothendieck the of formation by a commutative ring to pleted -. (j) identity element with Bj = e multiplication being automatically. along carried respect to the addition, = {f.). n(j) that and j J V1I be E w an that ring space can Analogously, chapter in see shall we Lemma 2.3. respect with pair operad to the maps is ) ,a)? (a an which is f completed to an E w ring space grouplike with respect to addition, by determined B -, XZ. ... A:ZkXB.X B J~ J jl---jk Indeed. structure carried multiplicative Ew the being automatically. along noa this topological the algebraic completion on completion will induce consider- of implies ring space the existence The 1.3. E an of definition and the equivariance formulas (c) and (cl) of Definition h w F The follow- a space. such of groups homotopy able the higher structure on Indeed, the equivariance formulas are dictated by this lemma, which g obvious Ing lemma displays the most bit of structure. with asserts unit and the distributivity that compatible are these formulas P - Definition 1.6. formulas of "pseudo (m a topological is )-space ,m An be a Let i' Lemma 2.5. (X, 6, c) (C, fi h- and where )-space, 0 groups homotopy the Then n*X defined connected operads. locally are semi-ring", "pseudo" commutative meaning that the addition need not be k- L respect to with graded semi-ring; a commutative form zero basepoint the be additive inverses. and "semi" meaning that there need not C is if n X / a ring. a group under addition, then T*X is Lemn~a b , (n , ) operad any For is 2.4. operad pair an with 0 f- f Addition in by the additive product B2(c) n*X induced is Proof. the only maps respect to possible namely A, [ the o c any [or h(2) go any for $2(g) multiplicative product Since c(2). b. all (k). for 8 o g fj A(g; f. . f. , ) = . . , E- ~~.--j~ J~ 1 pairings induces it XAX, smash factors through the product l: L i (n ,an An tn )-space is a topological semi-ring and an ,n)-space and : of the product letting by jl0 i, all n riijX + @njX riX S - X for The is a commutative topological semi-ring. unique maps + n and p : Sj -, X be the composite F .b operad fi -*n define a morphism (c , ) - (n ,n) of pairs for any pair operad is ). (t Thus any , commutative topological semi-ring R PJ ! pairs operad E it will exist, As soon as we that verify Definitions from directly follow semi-ring a commutative for axioms The a )-space. & , ($ t w and 1.6 f are k(j) and 1.8 and the assumption that the &(j) connected Conversely, the set n X R is an that E ring space. any follow such 0 w (compare p. [45, 41). L commutative be a will X space ring E semi-ring an of components of CO one categorical E suited The simplest and the pair, operad m to and the E of map discretization be a will map n -+ ring spaces. X X L 0 w k- from product- the of application by .a))) derived is applications, (a'/i & would be b semi-ring space Evidently, E a technically accurate more m I- spaces to given by contractible spaces preserving functor ID*(?)] from 1, term E ring space. than w [45,15.1 this of Recall from that application and 10.31 and 11.11. [45,10.2 5 i

78 Q(j) functor an E operad such'that gives is just the normalized ')?') to bar constructions are specified by [45,9.6 and llcre 11.1 (see p. 126)], the m Z. -bundle, universal Milnor's of version and both maps are of -space morphisms is D, X) X C Il(D, a J a strong deformation is &(e.rrl) retraction [45,12.2], by Q-spaces X Lemma 2.6. of 1 (Q action . & ) is an Em operad pair: the on inverse with right and [45,9.8 by ) t(q is a 1,l) B(rr2, and 11.10], itself is obtained by application of the functor on 1 the action of to 3-Q ID*(?) 4(ii)]. A. 2 (ii) and A. [46, by equivalence l~omotopy itself. com- as The formulas of Definition 1.6 can be written out Proof. as let = in (iii), let GX OBB(D, C X D, X) and X For - (IV) hence, mutative diagrams, by functoriality, hold for these formulas = fi c0B(rrZ, 1, I) 0 com- group a natural is g (ii), By GX. .-+ X : (q) T an. for so do they since pletion of the existence The such a construction was X. c-space of given there was incomplete. but I], 2. [46, in :~sserted the argument A description of (&, f& categorical be given in section 4. will The 3. VII § needed in be will remarks following I claimed that in In the second result labelled Theorem 3.7 [46], (v) assertion That clearly is a morphism of f&-spaces. on DX (It was Zj operads of an inclusion there results and = Do(Zj), 2.7. (i) Remarks as mistake commutative. The actually is CB imply that it would lirlse, pairs and an inclusion of operad (a)l,m ) C (Q, Q. fd. c m Thus a 761, [46, y a factor p. from which in {kccurs formula for the product (with monoid also a-space (Q, a and a topological CB) Q-space is 4). . . , j k ) was omitted (compare section rr(jl,. is also a topological pseudo semi-ring (with second product The pro- 8). 3.1, we shall ignore of (iv) in the the use from Aside proof of VLI. ducts terms given those with in O and CB coincide of the actions as (vi) from spaces which result the classifying structures on 6). -spaces monoid and (e2). e2(e2) 6 2 of (Q, application by constructed deloopings the favor in &)-spaces et\d and Ligaard Madsen [39,2.2] have verified that any Y Q.-space (ii) special have numerous latter The the chapter of machinery VII. commutative H-space with a strongly homotopy to the pro- is respect are proven not I have theory, and our two to essential properties that the the classifying an the and H-space @. Therefore is space BY duct I quivalent. n natural map sense completion the in [46, 5 : Y .-+ OBY is a group of [47,15.1]). 2.11 (e.g., by bipermutative categories Symmetric and bimonoidal 5 3. replace general convenient to It is sometimes E by spaces (iii) m Given be equivalent -spaces. This can done the as follows. with very provide a internal structure appropriate Categories Here categories all E is an & where 81, (X, -space & the maps operad, construct m spaces and E a, ring spaces. of source rich E a, and topological, with are to be small structure and all functors internal a category For , a to be continuous. are transformations natural icrid

79 @a of morphisms of a and spaces the denote &@ and objects and source, target, denote C and I, T, S, identity, and composition func- the continuous. required to be are which of all tions, If no topology in is can impose the we sight, discrete always topology. category is Recall that a symmetric monoidal a category to- FADE* and FAnFB FBnFA FA a functor with gether ~xQ. - & and 0 * such that t] is : an object associative, (right) unital, and commutative up natural iso- to coherent permutative if U is § VII, 1 and 8 71. a is [40, morphisms a, b, and c no isomorphisms required. strictly associative and unital, with Coherence - -+ a F: a functor a' CL' of is permutative categories A morphism the commutativity c, of isomorphism piece the remaining with structure, FA on B). F(A = FB 0 u F* B), and Fc = c * m F(A = = FB o FA , that nuch then guaranteed commutativity of the following diagrams for is by per- categories monoidal Note that a morphism of between symmetric c E @a. : A, B, of a morphism be permutative categories. not need categories mutative elaboration of the proof of slight A 4. following 21 gives the [46, Inore precise result. There is a functor @ the category from of 2. 3. Proposition can categories Symmetric monoidal be replaced functorially by the category permutative categories of to categories monoidal uyrnmetric but categories, equivalent permutative naturally of notions relevant the symmetric categories. monoidal of equivalence a natural a irnd - QR n: morphism require explanation. This is particularly so since the usual is If permutative, then .rl is a morphism of permutative categories. a categorical of definition a coherent functor between symmetric symmetric monoidal two with categories encounters often One would allow forgetful functor from monoidal categories examples like the additive one ntructures, and the (right) satisfy which multiplicative, one ring modules over a commutative R under groups under Abelian to @JR distributive nullity of and zero laws up to coherent natural isomorphisms aZ and too lax for our purposes. is say symmetric is a category that such shall We bimonoidal. n. and d symmetric A morphism + 0' monoidal of Definition 3.1. particular, In [35] categories. such of study a careful has made Laplaza categories is a functor F: a + a' = a with together * * F such that diagrams the commutativity of ensures a list which that he has given of natural isomorphism $:FAD FB -, F(AoB) such that the following dia- diagrams ti11 further coherence which can reasonably be expected to commutative: grams are of notion with the List Comparison of his commute. fact in do commute

80 155 Definition 3.3. bipermutative category 8,0, c,@ ,1, ?) is ( A a, 3.4. A morphism (Z .-, @' of symmetric bimonoidal Definition permutative categories of a pair @, CB , 0, c ) and ( 1, E) such (a, a, F: to- = n and 0 = FO such that @ &-- 1 a functor is degories Fi satisfied. are conditions three following that the and F(A - FB CB $:FA B) fB gether with natural isomorphisms : ,It of morphisrns are ) + (F, and $) (F, that such B) @ F (A .-, @FB FA o E ~@I(o) = I(O) = I(o)@~ and 64. A for A@O = for = @A o (i) nymmetric monoidal categories commutative: are diagrams following the and @. object zero two-sided a strict is 0 for is, that ; JMCi E f d satisfied by objects and The right distributive law is strictly morph- (ii) CB @FC) (FA i 0 " FA &FO and (FA CB (FB@FC) FB)@FC - I isms, and the diagram commutes for A, B,C e @LL : following -C Cc Ita is a functor F: a bipermutative categories k* of morphism A distributivity isomorphism Define left a natural the following as 1 (iii) permutative categories respect is a morphism of wl~ich with both to the composite symmetric Again, a morphism of and ntlditive multiplicative structures. \~lnionoidal bipermutative categories need not be a between categories diagram commutes for A, then the following B, E D C, : bipermutative categories. of rphism IIIIJ Proposition 3. from There is the category rn of a functor 5. bimonoidal myrnmetric cate- bipermutative of category to the categories gories and a natural equivalence r: GCL---t a of symmetric bimonoidal of a morphism is r then bipermutative, is & bipermu- .~tegories. If t tiltive categories. assume either To avoid technical topological difficulties, Proof. implies that a bipermutative work category is Laplazaas [35, p. 401 or are 1 and 0 that practice) case in the is (which discrete is tlrikt bimonoidal. symmetric In commutativity, strict of the absence clearly is it i and CB for unit a strict that is is oon-degenerate basepoints such 0 a strictly. unreasonable to demand that both distributive laws hold The by growing arranged be can condition latter The always @ . for unit nlrict law to make strict is of logically arbitrary, but our which choice choice the category with 3 (by adjoining copies of 0 and 1 whkskers on the given in the by lexicographic ordering used consistency dictated with is 1 and 0 two objects and one non-identity morphism) so as to obtain a new Notations 4. 1.

81 ~l~~~'llli~,,~~~3!,~ I 151 N r as free required. We construct Let (Oa)' be the = as follows. n(B@C)L-rrBCe.rrC TCCenB --=-n(C12/B) product denoted by W monoid, (4-a with " , generated by topological C! ~C@?r~dn(C[ii;l~) 2 -L~B@~C T(B~C) $i A the relations modulo for EF& all . Let 0 A O= = A B 0 and i = e ; isomorphisms objects B and C of the uniquely unlabelled are be IS) the free topological a with product denoted by monoid, , generated El Define q : ma m4 - of a . armined by the monoidal structures by Extend = e relation the modulo (@a)1 product the from 0. bil f, by m& -t 3nbJ T: and define A1 zl(f) = (A, -t A') for f: A ((Xi)' the formula by (963 of all to TB T tl, g, B') = functors, are g and q Then . rBr - for g: Bn) (Ai@ ." ' Am) Fl Q "' El (Bi & ~TB) ITB, of (B, 11 the morphisms and is the identity functor, = (A~H B~) . (A,z.IB~) EB . e . . . e (A~~IB,) EB . qr B. of identity functor the and fine isomorphism between a natural . . . a (A~UB,) straightforward. lo remaining verifications are equally B. E Ai. for . ((SO-)' on products associative, Both are 0 is a Cf@ J a strict unit for a , unit for Ell and zero for , i strict is and categories Bipermutative E ring spaces , Let and the right distributive law holds. denote evident q : 6CL + 6% the 03 inclusion. Define T: 08 by i, n(0) 0, a(i) = = categorically, (Q ,&) operad pair E We here describe the m -spaces 4! obtained the to categories permutative from iew passage ) )) ... (A,-~@A~) ... &.@(A~@ A~@ = T(A~~.. . m~ n of [46, $41, and construct a functor from the category bipermutative # and i, Ai and 0 # Ai , 00- E Ai for to ories the category of ((a, &)-spaces. dA;m BAS) n = TA~~O(TA~O(TA;Q... (BA;-~@TA;) ...)) - ... Recall objects has G monoid a of G the translation category that by aY16a Define 0. # Af (U&)I, E Af for from and morphisms G Ilr~! of elements G E g those elements g" to g' g uch g". When G is a group, = is unique and a functor with glg that = {B] X ~(TB, a(B, TB') X {B1] . B') 4 Note object its by determined function. is G uniquely therefore rirnge and S determine sets singleton The induced are and C I and , dj for T ,., G on the right from acts t11:rt G G. of product the via the corresponding functions for from h . a fi is topologized as sub- - .., defined functor be the -2 ~et ~:z~xz. x... xS. . the QRX sym- and on @ and products The 'maXQaj space of ;- J i J~ ji+...+jk [- objects by the formula 1111 determined : of 2' of 83 are arrows by the following metries and c - - N - @T jk). ,..., )r(ji ?(n;-ri.. @... (7 = ,rk) . . (1) O TC vB1 O TCI - r(B1 Cr) T(B C) --=+ TB (k) ~-~(i) u- from omitted inadvertently u(ji,. the definition , . was . jk) factor ('f'he [46, in given ) 821. p. of fi of C1) g, B1) (C, and and for f, (B, morphisms

82 N- .., - ,.d XZ. Let the functor defined on X... X:ZkXZ. + Z. be Ji Jk Jie"jk the formula objects by xc. x... ixc. . Let the classifying ~a of space a (small, denote topological) -~XLL k X.5. X dk 1 Jk category from and recall that B is a product-preserving @.. functor 'k of isomorphism categories. evident shuffle the is p ~11~:re coin- [46,4.7], in observed As BE [46,4.6]). g. (e. to categories spaces (1) to By of comparison for any topological G. JD*G] group cides with 1.3 gives following the of Comparison Definition to lemma the maps y the 2, of the structural formulas in the equivariance 1. Definition I (ttiuequence. (r;! operad maps the with Em coincide 0-j) - BCL. B(%.x = 80) x 4. Proposition BC.: = 8. Define Z. JJ J of lllrn the 8. define an action 8 a on Ba , and B restricts to a J from permutative categories to 'gL -spaces. ittnctor Definition 1.6, in formulas the equivariance to (2) the of comparison By '45. with maps coincide itself on the of the action give which X maps Then let Now 0, CB, a bipermutative (a, c, @, 1, '6) category. be of implies cbl~t.rence 4 following 4. I. analog Lemma the 0, j 4.3. The following diagram Lemma 2 is all for commutative operad pair be the define used to Alternatively, this description can E OJ j: = jk > j. and 0, '- I* . jl.. that such 0 I - (Q,Q). a permutative , * , c) Let Cl category. As pointed out in (a, be p. 811, c determines Z.-equivariant functors 146, J CJ on {e.} X aJ = aJ. of iterate j-fold the to restricts that c such J j section result. the following imply previous the of diagrams coherence Th: Indeed, the very meaning of coherence, we need only observe that the by *hel.c and formula the morphisms by objects on defined is w objects. of lemma diagram the on makes sense x (U,T~, = yk) ,T~~ . . yi,. w(r,~~, ,T i. . . . Lemma 4. Y~)). diagram The following 1 is commutative for all 0, j \ i ... . T~sfi k' that j = jk + . . . such + jl : k 0 2 and j. 0, 2

83 ill I/ lll1:'i' 11 I I '11 81 1 l,l,l'l ,' ' I tbl 160 their 2 0 and n Let 6 finite sets denote the category of and :7 1 the 5 gives 1. Definitions to lemma Comparison of the n c'(n,n) ,n) . rnorphisms. {1,2,. as and identify with of Think . following consequence. 1, c,@, 0, d, , (6 Then 2 group a bipermutative ?- is symmetric 8 on a of g and B& specified actions The 4.4. Proposition E- n. morph- defined on objects are defined are and @ d and 8. by = Bc. and 5. = Bz. give of the E operad pair (a ,a ) an action JJ 3 m 3 from Ba on , and to categories bipermutative B a functor to restricts (&,a)-spaces. (5 and fB is The following useful. sometimes addendum let a i e , of a. denote the subcategory 4.5. For Remarks object unique contains the which ai to i morphisms from all and B i i; and (~63 Clearly monoid of of morphisms. this classifying ordinary the is space is a sub a-space (B k0, 8) is a sub I& -space of (BCL, 0) and (B ai, 5) (BR,5). of l( ism nti if way There is a more general constructions above. of looking at the c(i) = m {D*Xjlj category simplicial spaces of sets, since the D X. are discrete). The (or 9J [45, $ill result by passage to geometric realization B& on a of actions g of L- p'or 8, mn should be thought n)] (m, , . . . 1), (m, , . . . n), (1, , . . . l), {(I, as L Although from actions of on the simplicial spaces B*a [46,4.6]. f- 1.4) and it is this choice of order (required for consistency with Notations a. no such examples will be studied volume, there exist simplicial in this - ! The leads to the strict right, rather than left, distributive law. which I - which are not of a* B_* Q a*) ( pair the by or spaces by actions with lipace r G)/Z:~ 9 spaces orbit the of union disjoint [45,8.11], is DS' the their clearly our theory will apply to for any a ; B*& the form i of with the disjoint coincides Definition 2.1 SLI~ unit e: DS' -c BE the realizations. the homeomorphisms of union !< k k i j Examples of bipermutative categories § 5. EX use (Alternatively, [45,10.3]. in ttpecified /: therefore We BZ..) = j~ seminal into map the will which example, all The is others, is the free (41, a)- and conclude that BE i~ogard e as an identification ti one. following C 0 & space generated by S . e b

84 For the K.A for i > 0 can be defined A, a topological ring groups permutative categories in terms of or finitely generated @A of 3.A and c clcterminant one, since define permutation matrices required to the 5 1). A-modules will be discussed in VIII (as or projective free left A When permutative category can we present, obtain a sub be no would longer I commutative, is be taken bipermutative PA and .3A can categories. as of numbers & under restrict- and the even to objects restricting by CB case the In of be more explicit. can we YA, itl~ define Similarly, A). SGL(Zn, of the elements to morphisms -f 4l-J.f. = 0 n Jf' follows. the $7~ 5.2. objects The are as .Q~A Define of Example n n-matrices X A of set the above, examples the In entries in with n as A together with its non-negative integers, each thought of standard and topology obvious product the given be and O(n, A) are -0 A) GL(n, ordered basis The morphisms of are %XA iso- the en) . . . . {el,. n n that insisted rings be have We 1,. subspace topologies. IIL. given the . and m n if -. A is Thus J,x~(m, n) empty morphisms A simul- topological K-theory and algebraic treat to order in c.~l*~~logized by 6 -c /!.$A a functor Define e: on e(n) = n n) = GL(n, A). 2 xA(n, IIIII~~~us~~. objects and morphisms Then = e on e(u)(e.) u e I: . ) n r-l(i ' is A CB, 0, c) /2J&l, a permutative category and, commutative, is if IR, ( i.r.~mple 5. 4. Q;, and H be the (topologized) real numbers, Let b tttriplex numbers, and Define subcategories + = i? 6, % quaternions. 0, and 2 c,@ , is a bipermutative XA,@, category, where CB,@ l,q , c, ( f that requirements the by specified are usual meanings their have @ and % fi to orthogonal, unitary b4Ii-I restricting of by ZIR, and JF xa;, fJ nt~l - CB Am respect to isomorphisms the (with Amn -. A~@A~ and Then are and.'u @ ~ntl respectively. linear transformations, sy-rnplectic :f defined as usual ordered bases) on and that a morphism be $A fJ + of I~tp~.rrnutative a morphism of categories, and complexification 0 -- IL is & e: Note that categories. commutative, bipermutative is A if and, permutative i~ii~vrmutative categories. 4 is an (additive) permutative category and -1 the naturality of the unit of factors through any A. By e for 21 r,y~r~plectification permutative categories. of a morphism is 4 + Be: (&, Q)-spaces, BE: -L B!J&'A coincides (under the identification the forgetful functors 4 -. -& t'f111.1l bases, specified on appropriately 0 DS = B & ) the unit of B b YA. We identify morphisms of 8 4A with %L with categories, (additive) perrnutative of morphisms are J~IIII 19- + respect with their matrices with bases. standard to the send n II~IJC-T~ which functions 2n. to For a morphism of f rings a: define A-~A: entries -. to a applying by XA' all 9 of ~JXA bqa: significance The three examples, whose following was first under- matrices. Then from rings to perrnutative categories and hx is a functor sc6,ucl Quillen interplay to the central are [75,77], Tornehave and [58,59] by from commutative rings to bipermutative categories. and topological K-theory VIII. lintwcen to be discussed in chapter algebraic For of characteristic a perfect field k Example commutative, the sub be bipermutative 5.3. If A is to define OA .xn~,~ple 5. 5. let q# 0, matrices morphisms are the orthogonal of category .uqA whose $$ , and &k+@k dq: &;ek -- MYk t also a is a bipermuta- rings ative commk from functor to Then (MM I). = tlnnt~~e the Frobenius of bipermutative categories derived from the morphisms

85 IllI,! 1 bz automorphism - x of . xq k be the a-fold iterate of ,dr let qa, = r For The O(n, ) det( u ) = 1 for u E Z C u k 3). ( dues and v ZZ in n This when most interesting is example the algebraic closure is k of $q. n egory 77- kg those are n + of morphisms whose @k 3 of field the q elements. (~)det(r) v kit category bipermutative a sub is 1 = a prime be q where qa = r Let is 1. and a 2 kr Example Let 5.6. formulas by given are sums and v direct on same form the of det e with a field a forgetful functor elements. r Define &xkr-+ f: E 5.6 works equally well with &Xk tensor products. Again, Example by 3 T and letting f(~) be m regarded as a permuta- = letting on objects f(n) set kn tion of the ). GL(~, T E k letters morphisms rn of on Of course, chapter examples to shall We have only listed refer in which we n from sets of isomorphism chosen the on f depends to k: . . , r . 1,2, . and examples, our o£ many by Swan [unpublished], pointed As all out With the lexicographic choice, obvious of morphism exponential an f gives within a groups can be subsumed rs, general framework of systems of permutative categories together for n 2 0 + G(n) X G(m) G(n), - Zn homomorphisms with CB, + ( I,%). fi4kr, f: 0, C) ( ,a tr tn), for the bipermutative case, and, G(m) X G(n) -+ G(rnn) subject any to which apply remarks, The axioms. appropriate he following of morphism Moreover, the composite @-spaces h example, describe the action maps BG(~)' BG(pn) and X, - ep: p(p) XZ 6 -, BG(~') p: Q(p) BG(~)' P P 2.2 since B(fe) e unit exponential the with coincides defined in Definition 0 a prime, .p the is When of groups. homomorphisms of terms in lely sends to 1 This 1 to 0 . DS of &.(r)/Z component in the a point and maps uced of mod p homology determine operations on fi.2;kr replaced by works equally well with @'kr. Z BG(n): fi ( thus is operations these of computation the and 1, I [26, ) consists Example 5.7. Let k be a field of characteristic # 2. O(n, k) P n20 to educed representations. appropriate of analysis homological the isometries of the the to associated B form bilinear the to respect with 2 Let objects with category permutative be a c) 0, (&,@, C2: kn -, quadratic standard map k , Q(xl,. . . , x J = Zx. 841. p. [51, n 0. from morphisms with and a topological group forming n to n norm Recall [51, p. 1371 that the from 01 I n 2 spinor v : -. O(n,k) k/( is k)' all that the wreath product Z JG(~) is the defined by P ... T k). O(n, T=T if R(zr) ... Q(zi) = V(T) E determined ~(n)' and action the evident by of Z of product mi-direct P 'r '1 n regard G(n) with a single ~(n)'. If we categories Z and (~(n) on as Here - 0. T (x) = is T Every x # y with k E y x, for Y)/Q(Y)]~ [2B(x, P Y ," of ~(n)' X ~(n)'. 2 rbit category X X and, modulo such of a product is v (T) squares, 1021 p. [51, symmetries P Z~ G(pn) + ~(n)' X homomorphism the through factors he functor c : X e. - per- = T then e y the choice independent of of If factorization. Y j' 1 k to specialize NOW 2. = ) V(T and e e. mutes and Then v k3. = j Y

86 167 t Z) to be the sub BO-space BO(0, z)ABo(~, the of BbZ. D(BT ) is 0, free t t of (Q ,a)-space generated by BIT , and there results a map D(BT )-c B6Z Application space functor B thus gives the commutative of classifying the is 2.43 [45, (compare just the identification (i)J map fact, this In ,'&)-spaces. diagram t BO(~, : ) E .L!. a (p) Xz B0(1, z)~ ----z .L/. 8 Z) = B@ Z. D(BT &. P t injection e: + 82 gives rise i'lhc the $ DS' + D(Bv ) functor under B to + 1 and 0 points the by cintcrmined . BIT of F) If, 0, c,@, 1, (a,%, is bipermutative, then the functor further, : X ~(n)' - factors through the homomorphism zpfG(n) + G(np) ~(n)' P P specified by (~;g~,~..,g~) + u (g @gp) B gives the commutative diagram of and application use these remarks determine to We following The BEZ. observations Z. to due are Fiedorowicz. an for convince reflection moment's A that, will the reader 5.9. Remarks t row and column M each if and only if of I = M, integer valued MM matrix natural Indeed, the + 1. entry is one non-zero has precisely and that entry T: all p. Abbreviate (0(1, Z) -C O(p, Z) is an isomorphism for homomorphism P m and a disjoint basepoint O(1, union BIT', regard RP 2) the = IT of BIT r and

87 lll,l~~l~l~,~~l' I.ii Iil Ill! 169 equiva- is rX then l), rrlll X K( see union disjoint the is X if that U j' ring spaces recognition for E principle VII. The m 0 nt as an E ring space to RS . This result is a multiplicative elabora- m theorem n B arratt- the Ruillen of [16;68 746 $31. identity element of r component The the X of I'X is multipli- a 1 an is of space The zeroth a spectrum E space, E an and m m E 5 that, under space. We e mild section in prove hypotheses, the m theory. space determines a spectrum and therefore a cohomology If of I' X at any submonoid M of the positive integers is calization 1 E I'X the zeroth space denotes of the spectrum associated to an m as identity element the of component to the space infinite loop luivalent an a map there then X, is E L : the X -*- I'X which respects space m f the zeroth space of the spectrum derived from the multiplicative E a sense the in completion, a group is structure and that of on a words, subspace XM nce X. In other structure certain (Pontryagin) ring the a localization L*: H*(X; Q) + H*(I'X; 9) is of though I' x is use of the additive con space structure .strutted by E 1 m n submonoid its at k) every for X coefficient H*(X; commutative 0 space structure. E multiplicative depend only localizations its X, I onthe m I' of a discussion letter the definition; this ring X '. (See [46,fjl] for the localized only multiplicative to approaches earlier structures, the of a reminder has as chosen is and property completion group were handle no there visible because way to spaces loop >finite the was infinite letter other theories this of use the with do nothing to of in in the particular, In combination. tlditive structures multiplicative and spaces. loop the coordinate-free in understood to be are spectra Here ) paper of result of Tornehave's nin [76], which describes localizations and461 11, and the results of [45 just sum- sense in chapter introduced case a special as out by drop will groups, the of terms in symmetric be recast in terms of such spectra in section 3. marized will of IIIC version of the Barratt-Quillen theorem. our In chapter IV, Frank Ruinn, Nigel Ray, and I introduced the obtained of essential results this chapter The 1972 and in were E ring spectrum. In section such a of the zeroth space 2, notion an of m since this I mention iwc!sented 1973. of winter the in lectures during an E will be be shown to spectrum use requires The space. ring proof a author has since least nt one other announced his intention of developing operads convex bodies little the of the introduced in section 1; X v theory. L :I similar 5- essential feature of is that group the orthogonal OV it. on acts 6- E additive the by spectrum determined the 4, In section of structure cD operads bodies The little convex I. 5 will an E ring space be shown to be an E spectrum and will it ring be m m both E proven that, for an E ring space X, L :X - I'X respects The operad little cubes played $41 [45, role in cn of a canonical m m space structures. E In effect, this means that the multiplicative spectra. E spaces [45, tlie to from Indeed, as explained in passage m m preserved on E additive structure is the passage from to its structure a space 1'. 153-1551, the geometry given by the action of any E operad on m m associated spectrum. general results, we As a special case of more was automatically transformed into the little cubes geometry on the derived

88 171 - &V, and there is reason to E We a canonical need space. spaces such construct to been unable have 1 to obtain infinite loop order operad pair in m shall We be iorced to re- I~vlieve that no such spaces of embeddings exist. from the analogous (but considerably more delicate) passage E spaces ring m by the following weaker condition (4) place to E spectra spectra. From IV, the ring definition E of in chapter ring m m it is clear that the linear isometries operad of I. 1.2 must be chosen as CV is a contractible space. (4') for structure. multiplicative the operad the canonical operad an require We to conditions the then, I.:ven (l), (2) , closure incompatible, be and (3) appear and ym &_ which 7 acts on which can be used interchangeably with in to have shall we However, drop .trlcI shall be forced to we altogether. (3) operad, theory. one or the additive, of sequences of embeddings which satisfy analogs spaces I1.1ve composable and Recall and J* from 1.11 $ I. 1.8. Let 4' the definitions of in proceed we mind, to With these considerations (4'). and (Z), (I), ,,f : denote their respective full subcategories of positive dimensional and $ ~$11 r basic definitions. spaces. real product inner Ideally, k a functor construct to like would we inner product Definition 1 .l. Let V be a finite dimensional real of the category from 9' to operads such that application of x to R~, c: V -+ V convex ul1,tce. body A in V little is a topological embedding equivalent to Xn operad an yields (m, n 2 1 we In fact, . shall have ' n by ct: maps the that otrrli V V specified deal a good for settle to While less. Xm will be (weakly) equivalent to tm* n for m be) (or not be < will at least will not be proven to the V E x for (1 ct(x) t)c(x) f tx = - at all, kn. X functor the construct to order in Moreover, the to e quivalent that it follows = c I. Since E t embeddings also nrr all for c t,s stt-st' to carefully have shall operad and of notion the weaken to have shall we (c1, c A little sequence ) , . . . of body. convex a little again is at~~ch c t q in examine the resulting geometric structures order to make sure that inductively, if , tiv vex bodies is said to be composable if q= 1 or, > 1 q machinery of the [45 and461 still applies. 1111r1 ,t, c~-~ , . (c~,~, e < i . 5 1 c~,~), , . . . . c~+~,~, c~~~,~, c~,~~ quite simply. explained be can difficulties The carry out To our to original program, a space construct we have would EV embedd- of It follows bodies convex little of composable sequence .I t I. E for all in ings V V space product inner real each finite dimensional + V for c , . . (clat,. each I~I,I~ of use and, again composable is ) inductive by the q, t such that the following properties were satisfied: all sequences obtained by composing c of the maps some R~IV t = 0, that fcf-' a dimV=dimW. EW 4(V,W), fe and CV CE if (1) com$osable. t fixed) are blocks, ordered (m with r (2) cXd~&(~€B~)ifceEVandd~&~. the Although considerations, the convexity from definition evolved &V c,cIE ev. if c.c'€ (3) convex (1,l.n~ need a little of is convex body a misnomer: body the image j-tuples of space The of of elements with V disjoint pairwise & (4) convex. I,c iutr the configuration Z.-equivariant homotopy type of the images has J j) of j-tuples of of F(V, space V. points distinct -.

89 lIl,!'J\i!~ 11,') 173 Examples embedding which is an increasing 1.2. (i) is an If c: R R union V, of copies j of regard disjoint function, each is so then ct. Any (cl,. , c ) of increasing . sequence . subspace v(j) as a a map J~ + V, and topologize R -- composable. is embeddings R < JCv:(0) = > (Regard - 11 V. continuous functions 'V convex body and (V, If c: V - V is a little W), f a Q W, dim = dim V data The requisite of V.) set edding" the empty in (ii) 1 then (f c = of sequence a composable is ) f-l) (cl f c f- . If . . . , c 9 t tj . jit.. k ji j little convex bodies in V, then c f-') . . , elf-', f . (f is a cornposable k~= V+V ..+ Vt. +%): ,..., ... ~(c;d~ d)=co(dt ) k i q bodies W. convex little sequence of in ck> E c xv(k) and for = < ci.. those . . , V and - are little then W d: W + V c: If convex bodies, (iii) . . r,i, d (c pair that each such v(jr) )(, E > drIjr , . ) r' r, s (c (cl,.. are . X d)t = c X dt. If composable . ,C ) and (dl,. . , d ) a composable sequence; is t 9 9 then (c X dl,. and function; the identity is Xv(i) E 1 . . ,c X d sequences of little convex bodies in V and in W, ) b) 1 9 9 > for ua 22 W. a composable i,...,Cj> = C. , set . . ci, < those j-tuples of be the (j) )( little of . v 3 (R~, V dim V), maps the Then n. Ict f E = specified (j) gn(j) + v convex bodies such that the images of the c pairwise are disjoint.

90 175 -1 -1 n specified is k deformation the requisite Indeed, W. and V all by (a anc f the body convex c to little n-cube a little sending by f ) choosing and defining Xw(j) t 3 dj , . . . dl, < point any contains a kv. Thus Xv copy define a morphism of operads en f). en for each such pair (a, of . . ,cjXdj, homeomorphism increasing an Fix Lemma 1.5. Xl, ..., = c.Xl) k(c L 1 J morphism be Xn -C en : by specified operads of i the a: let R -C and J n . 1 5 t Xd.> 1/2 s , . C. , on . l~iblf because

91 0 The canonical E operad pair 2. m a space Define for letting by X space each X K show just as cn acts that naturally on n-fold loop We first v fi x'/(w), X, Xv(j) = "X where spaces. loop "V-fold" acts naturally on X so [45,5.1] We spaces j the which discuss the partial operads monads" then to "partial K x.) l,...,~i-l,~itls-..,xj) c.>,x itla..., ,..., (,xl a v J J J that 3( give rise. Finally, we show , K that SO xm acts on m q :X by KvX * (compare [45,2.4]). Define = ronever x. * restricts in monad a (partial) to VI. in (as [ye] < that prove and 1.9), would KvKVX [< : p define to like = We x]. 1 >, (x) by KVX an the zeroth space of any E ring spectrum is E ring space. m m Yit -. > ykl ~~1,. . . t Edk* yk1l = Y (c; dl* - - [ P[C; [dl# should begin by an action of notion the defining by a (partial) We jr is y since so do cannot We . c dr r X E yr and KV(jr), E E v(k), gC a space, of requisite leave the operad on shall instead but modification We there- should t . . . defined xKv(jr). on all of "Kv(k) X &(jl) X We reader. the 1.3 E term the use VI. to to continue Definition space w of a partial monad C, with define maps the notion [ore structural for with an action by a (partial) a space operad. With the appropriate E m GX The of G(~)X subspace a suitable CCX. - C(')X 1,: various for fix to tV as definition, the following result is trivial to verify. We agree specified p would have to be defined (and equal) on (n-l)-fold of iterates the that so V, of compactification one-point of be aubspaces required c%. (More precisely, c(~) would c(~)x .Zvx tV. nVx = F(~V,X) and =Xn n of C .) to the proper formal definitions, a partial be a subfunctor With tRn We agree to identify with t~~ = In/81n via the homeomorphism notions to gives rise a partial monad aperad that the way such a in C 1.5 Lemma in is as R -C J a: t(i~-l)~, where Here again, we of an and by C are equivalent. action of action an by and 6 $1 , E J let V X For 2.1. Theorem to continue and reader the details to the pedantic monad terms the use luave nVx ev,j: x ( nVxf * ,(j) be the map specified by x / only examples will be the The general more the for C-space hnd concepts. if V E b = c (a) yr(a) an (where $' is &I X of operads from G derived monads the form = c~,...,c.>,Y~,...,Y~)(~) .(< b/oijmcr if * J ev,J operad). Here the subs paces honest n-fold c(~)x of all are obvious: C% t r=l little of bodies iterates nequences the under composed be would which convex Then the of nVx. Under the natural xv on 0 action an define 6 v,j v to be composable required are p Definition sense the 1.1. of in of identification = nvbWx, the action aVSZWx 9 coincides with the com- v for have an action 6 : ~~fi + nVx We any space X. E j( 0 of 0 -, the actions , a: Xvbw. For .and E A posite of v vb v v w f KvX : a - to define , J* E V and gxvx Theorem 2.2. For X E define an c RW on action 8 Xv E I S2v~~ V for xm of E on v 0 m 0' composite the be I,. R~, Under the identification Q% = aVx, fi on gn of action the V = e Kv" , K~X - nVxVx : % * and coincides 0 of 2 the action with the composite of i n

92 where natural and define a : KmX - QX by the is L?vZv~ -* q:X inclusion, m m + Q are + SZvZv and . ' C Then aV: Kv R a V limits passage over to on gW ol R~ and is the identity whenever the orthogonal complement of _$ co'Km m k actions of K on SIvx morphisms of monads in 3- and the and of (R on the orthogonal complement of W C is ) . The cI the identity veri- v but elementary. 1.6 VI. are tedious, the E are induced by pullback along av and a from for E K on E [tcations specified in the identities of m 0 m Q~X~ Moreover, a and Q. of of actions a weak is equivalence homotopy by IV. 1.9, Em of a morphism is operads, then, m p : + then 9 is in genez al. completion connected and is X if a group 1, 5 VI in explained As u is a monad in b [ Te]. acts on if fi v v The monads Proof. [45, consistency in SZ 17 X and following as and are Q have the p. We defined a 1 Te]. monad in a (partial) is C 461, statement holds by slight elaborations of the purely the first and spectrum ring the which E any of zeroth space ~~catement, implies that m a = a with moreover, [45,5.2]; formal diagram chases the in of proof ring an space. 16 E n v m n V for + morphism of coincides with the nnxn monads de- Cn anin: , R = (Xm, 7) be morphism of a -+ ) Theorem 2 , & ( p): Let (a, 4. -* KmX 2(i)], A. [46, is a homology CmX im: rived By cited result. the in a ' - pairs. of a: C -C K 1.: operad Then the morphisms and Q Km ma m ln for isomorphism space any connected X. the last statement for Now X If E is a in to restrict tnonads 3 in monads of morphisms b [I [45,6.1], while the general case is proven by explicit homo- is given by a is ($ ,& pullback by )-space E zeroth fJ space its then -spectrum, 0 [26, f§ 51. in logical calculation . Eo a a along - QEo Q-action Jk B(~)x(Qx)~ - x...x~m(jk)~(QX) X -KmCil)x(Qx) H(k) I = { il, . . . , h} , then cI is the little > and c if where, c =

93 Ill,)' !Ii~l;llli] lo1 It also complexifies slightly since we must of points of QX as maps 6 is ed in the earlier versions. specified in VI. 1.8. Think on the left w Sm + x n Sm, where S cl, . < = c.> 6 , . = tRW. . For c operads account of the distinction partial and honest ake and their between and (j) w J the and constructions basic the However, re- the proofs of bulk actions. = (c, composite the then 9 y) (QX)', 6 y.) . . . . (y,. is y W.J J unchanged main not be repeated here. will and 6' a locally contractible (honest) operad, for example be Let [45 Xm x c' = $ that observe and 3.81 wedge map pinch the is F where 'Y denotes the Y, of j copies of n Define E operad. an or w by specified r * = Z(b) when a, and a some for c (a) = unless b = F(b) ia a (partial) E w operad (that is, Z. J acts freely on 6 and (j) (j) and the in the is p and SOD, of copy rth map. evident folding homology have their iterates and y trivial G For g the of domains the maps structural kJ (k) k a: , .$k(g. z) is that map which makes E follow- (QX) groups). Let zk) z = (el, . . . , the and projections. be the c1 -. C 6: and xw + For assume we technical reasons, $-space. ing Let 9) commute (X, a be diagriim fk Z A...AZ w a1 k once and for all that X 'is of one of the following three types (which sm ASw& ... AS ... XAS? AXAS - certainly examples of any interest). all include is X on of action The c' + : $ pullback along by obtained (1) on ' & of an from action X. as is of by pullback a C-space its regarded Eo Q-space X structure (2) (pg)t(V1@ . . . @ V ) where of orthogonal complement which, on the and k E . E where n, u along that is a finite-dimensional subspace z Vr C such RW is in the image w v v a C-space X is CY regarded as (3) via the structural map y of the the and X in has constant coordinate r~, r~ R as identity map of & C, . 3 e where Y monad (partial) SOD. coordinate in verified by Now the desired easily is commutativity g z55 direct computation, in the point the that the smash products used being In is of domain the (3), in CX; is 0 the the domain (2)' and (i) CI of in the used 5 distribute sums definition the wedge over definition of domain k . p of C(')Y of the . 0 of For V space product inner sub each finite dimensional R~, wj ?$ and X X c1 = define let in monad the be C $ to associated v ' v one operad recognition principle. The generalize of the context to and Recall, g- partial of a notion monads, the $3. 3.E of adjoint action (right) a functor a the 51, [45,9. monad.on By [45,9.4]. We translate the one operad, recognition additive, principle of v of gives RvZV K an action Kv : cr ol + By zV. functor the on v 21 46,s and 14 [45,§ coordinate-free spectra. of into the language It Cv of . X" on along an action obtain we K -.+ ,f! C a: slightly in the a prespectrum of the construction process since simplifies v v v' awkwardly passage from a prespectrum to a com- spectrum were and the

94 182 183 is the two-sided bar con- The basic geometric construction of [45] and finite dimension. V -c V' for subspaces V f: V' of Rm of the same struction [45,9.6 and 11.11 by cases the in arguments separate verified easily It is (I), that (3) and (Z), fv :Z + X such that if Z $(f):X -+. ZV' is defined in the maps there are J = B*(F, C,X) 1, C,X) B(F, (TX)(f) 1 Xf: cV -C cvl , then maps X as re- = Cf if and way obvious a monad, where C is X is a C-space, and F is a C-functor. Here I I specified be quired can by denotes the functor to spaces simplicia]. from spaces, geometric realization (TX)(~) B(B~,c~. E~)):B(B~,c~.xJ - ~(rlY', cvI.x) . = (6) the is X) C, (F, FC% where c q-simplices denotes of and space the B '3 most will appear c(f) details the since the omit We maps requisite generalizes the of This construction C. of iterate q-fold readily context to 5 would suggest) letter the of use (as theory operad two the in naturally partial monads and their actions. In the of practice, of being X to due one any in (TX)(f) no play case and terms C in the C of above and to the types specified definition three 11.1.10, the as and explained since, in v role. essential of q' 4( ,,, there , Km, and the will always subspaces obvious be m R u) is a prespectrum. Consider the spectrum TX (TX, Thus that B&F, X) FC% in sight so C, the appropriate faces and degeneracies of m natural the and map : by TX R -, TX of L 11.1.4 and given prespectra Indeed, requirement the to amount simply will this that defined. are S(X, 8) when clarity, denote the necessary for or II. rX, 1.5. Let the sense little convex sequences the composable precisely bodies (in of Q~TX. the is the recognition principle of crux The zeroth space of of 1.1) composed. be allowed to are Definition th analysis the zero map of L :X TX. - by thus define a space (TX)(V) We may diagram: the Consider following Theorem 3.1. X) cV, B(z~, = (TX)(V) I . I X) cV, B*(z~. (4) v v identity the are Cv = (03, Z ,a all Kv, and , V when convention, By th functor on a the and 3 zero the Thus identity maps. are p and v v space = (TX){O] is just X. T~X an orthogonal pair of finite dimensional subspaces V and W For ~(q); inverse right with retraction deformation a strong (9) E is (i) operads Rm, the morphism of of induces a morphism of K v --Kv+w w, 1,l) is a group completion and is therefore a weak homotopy (ii) B(a -+ gvtW first . 12.11, and [45,9.7 by given equality the With operads m e., is X n if a group); grouplike is X if equivalence (i. obtain an inclusion we therefore 0 is a weak homotopy equivalence; ym (iii) : Z~B(B', cV,x) = B(z~~~, .+ B(Z~'~, chW, X) . cV,x) (5) m 1, group completion. (q), 0-r 1) is B(amw. hence a L (iv) L = y I as u) gives defined a prespectrum, (TX, say that We would like to defined on 11.1.1. For this, T must be appropriately isometries in

95 IlI.1 lllll ~11'1 185 Formal results from the in apply equally well [45] con- Proof. (3). By suffices show to it [26,1$4], that in case )(qti)(~) K X - in monads, and monads and 11.10] partial as that of text of [45,9.2,9.8 - Cm)'(x) (C'X K_)['(x) X (Ct [45] which spaces also general simplicia1 apply to in Results (i). imply of sides Typical points have both an isomorphism on duces homology. [45, 12.3 and 14.4 (iii)] imply (iii). here, and apply equally well The first of and in various of the spaces ordinates the relevant operads in X or Y. calculation (iv) (compare of is a trivial part [45,14.4 last the and (iv)]), The successive filter by the of coordinates in X or Y. number may partial our by necessitated statement is proven by a slight elaboration, as thought generalized equivariant half-smash products tients may of be 12.41. 2 and [45,i2. structures, of proofs of the to prove It (ii). remains morphism any by spaces such on induced map the ,2.5], and of little have to use infinite operad cubes shall Here we the re- and we 03' Indeed, the shuffle artial) isomorphism. a homology is operads E w C and C' let X write C = C' X Kw be the monad associated to w chains spaces depends only on the p shows that the homology of such obtain the commutative : * of we i use By c,. X 6' 'K,, w latter The chains are and the chains of the Y operad coordinate. or X diagram appropriate of free over the and configuration symmetric groups, yclic con- standard techniques of homological algebra apply to yield the ce m pair of results show ,that The following S2 TX gives the "right" (2) and (3). ectrum in cases the Then E .a. Let 3.2. Proljosition e maps regarded as a is X where C' X C i Xiw. -space by pullback along By w , EV e(PIv): E~) B(X~,C,, = (TE~)(v) Xiw, B(i Xi i triangle, the left is [46,2.3], 1) v v a homotopy equivalence. By w' here EV bv: Z Z Z~E~ natural define a map, evaluation the is EV - S2 w i] in the proof 2.7 given there, the VI. with but (iv) substituted for [46., 2. map The - TEO ;:a map unique prespectra. of VE - TED of g. w: E suffices therefore ~(a i T, a group completion. It i, 1) is to top arrow 03 w that spectra such ( v ij) L = 0. induces an isomorphism on T. for all i 1 w Xiw, 1 B(1, that prove on homology. an isomorphism 1) induces By [46, A. 41, 1261, p. and [45,9.2 in defined is &(bV) state- first and the Proof. show to suffices it that each and II. definition, the 1.1, from verification easy an is ment (4) through (6) B (i,iXiw,i):~q(Q,C'~Cw,X) + B~(Q,C'XK~,X) and map the identity is w: Eo = (TEO){O} - Eo Eo Since of prespectra. q homo- implies that is Eo - theorem the of (iv) grouplike, is fEo .;,: a weak an isomorphism and (Q,C'XC~,X) B on homology. induces 9 follows. statement second The equivalence. topy B (Q, ClX Kw,X) are obtained by application of the functor Q to 9 the appropriate (C' and to C_)~(X) x (Cl latter the where Kw)[d(X), X is an stable the in isomorphism becomes u The proposition implies that f space, namely domain (2) )('(x) and in cases (i) and (C'XK thus E (and other words, HA if E is connective. In category homotopy m

96 the can be recovered from the underlying cohomology theory it determines) statement, which was used in we have the licular, following consistency details since some give We C-space E Bott chapter I. of discussion llle periodicity in ' 0 proven result there was more precise than the result needed in [45 and the Proposition 3.3.. spectra , e of composite map 5 the For Y 461. a map is there Then &'-space. a be Let X 3.4. Proposition the that such diagram commutes following d H in -CZdCZmT~ &:R~Ts~% deformation retraction. a strong is Proof. The zVy. = maps Recall that (zmy)(v) by strong [45, 9.9 and 11. lo]. are deformation retractions evi- the With f [46,3. gives maps of I] -spaces & Proof. maps dent B(1, Cf) = cr , as in (5) and (6), the B(I3 Cf, and c) cr, - 6 of CV, are the B(zv, CVY) a prespectrum TICY, and the maps spaces Bdn% x YdX - a strong of E(PV ) define prespectra. deformation retraction is There d E are weak equivalences. Think of as the Bdn% and 6 that "ueh passage TCY and, since obvious to spectra an inclusion of in is T'CY coordinatized the of space .croth . {Bdlin%) = S2% m SZ-d~ spectrum V over a limit process C R~, this inclusion becomes an isomorphism d gives 3.2 Proposition of and use m B the of application Then functor the upon application of functor am. resulting The retract- deformation spectra coordinatired of itlaps tion the is pv since specified composite the is QmY -- Q~TCY composite zvlr zVcuv 8, zVcvy zVy. - zvnVzvy - - I~~K~Y loop spaces between The proposition gives an equivalence of infinite space level the zerob" On which the first weak equivalences. of two are and RY and I'CY the of version preferred our is Barratt-Ruillen wc have the following commutative diagram of weak equivalences: theorem. 8 and further discussion and [45, 31 contain 14 2 8 [46, and 151 and about the coordinatized spectrum B X various additional results m (SZm~~)(~l). = by B.X specified used Of course, the little cubes operads replaced in those papers could operads bodies convex little the by be the intermediate space Y d X of Inspection of the explicit construction change in the introduced or their here without any proofs. results In par- diagram in p. 148-151 (especially the bottom [45, on p. 150)] and use of

97 d d -1 composite ($2 E)(n 6) L is equal in ~4.91 [45 demonstrates that the composites from induced will be maps R&O identity map a%. the to .? H of 4 functor the of application Now with the equivalence $2) maps commutes (which the of to and use of (*) coordinate-free spectra. of 11.1.8 JI$ ci 1 of gives the required map f; explained As in 75)], the previous result implies the [46,3.7 (p. following further consistency statement. Proposition 3.5. Let G be a in and 1. tf'[$ monoid Then BG spaces. infinite as equivalent are delooping B G the loop 1 the 51, [45,11. of the natural homeomorphism is arrow first the &re The two operad recognition principle fi 4. smash in simplicia1 each products to passage by derived is arrow icond and is of realization the third arrow igree, the a map Xj(g)* of contractible Assume given a locally ), (GI, operadpair for h' It is apparent from Lemma 2.3 tt~plicial spaces still to be constructed. example locally contractible any E where is ,& 8') or (n, operad "Li 03 3.- Bhl the maps operad pair ( 6 , $j ) to be the product (partial) pair, and define * g- (i .-.i) Xxm(il) (i.) -+ 1/( X... X k h:X(j) x+ (C ) -+ (x,.X) be the projection . Let (n, P): t!j A'x~ ). (&"x K,. m1 j mj of and regard elements via isometries linear as (j) b Recall the p. limits obtained by passage maps from to and adjoints ?Q & kf -prespectrum. a definition, IV. 1.1, of 5 3 ... Xxv.(ij) -. XW(il-..i), W= l(Vl@ ... @v.), r(m):?C, (il)X g- j J 1 J -- Let a (I;, &)-space. Then be 8,c) (X, 4.1. Theorem structure E ( product r the (j). and 1.8 VI. Therefore, in on view of 8) admits T(X, = a natural structure of TX -prespectrum, hence !d 5 .(g) such that the following dia- arrows dotted unique are 3J), U, there &-spectrum. Q~TX admits a natural structure of J gf 3 1.8 (the commute grams VI. by given being c.(g) arrows solid and 1.9): Proof. the IV. the first. from follow second clause will By 2.3, - - J 00 L and be (j, i 15 product Vi, R of sub inner dimensional a finite Let XCX . CXX.. : XCvX Cv XX... 1 j 1 g let .~ (j). We must specify appropriate maps c: j(g) 5 i j(g) e - G I J W where . .@V.). g(V1@. = (*) j(g): . .A (TX)(V~) + (TX)(W), (TX)(Vl)~. &hls statement holds for any /jO-space X and in particular for CX, F ;"- map in role view of the original played In VI. 1.8 by the ~CX, etc.

98 191 5 we see that iterative application of the statement above j(g):~J --+X, diagram chase easy An 5 in .(g). functors the of definition the employed J arrows unique yields dotted 5 .(g) compatibility the that rtl~ows between the l(e) 1. IV. condition such LO, q dia- following that the 5 j(g) and S' J grams commute: Finally, 1. IV. condition l(f) (Irc maps cr is 5) (3. formula of satisfied. map holds if we define the ol;viously for 5 (f):X * X required formula C cqxx c: xx... XC9x ... xc; - X j 1 I I gl~ = f. any for (g) &(I) E such that g 5 be to (1.6) 1 know spectra. We from ring Thus Em ring spaces determine E m how the derived additive structure (that is, the spectrum i.lirorem 3.3 next given space structure. E additive We the related to is ~~ri~cture) m all pass to 5 smash products. If we collect the smash product The .(g) to structure multiplicative the derived pclate the given multiplicative J 9 . . . , tV. together and apply tg to them, then we obtain from factors tV1, at ructure. J .(g) 5 the further maps the Theorem 4.2. For a f$ (& , )-space (X, 0, E), all of the maps J q m 3.1 Theorem in specified L and ), maps , y l), 1, B(ums. t(0), ~(q are are the first three of these maps oi maps of O-spaces, hence b B~(x~,c~,x) maps those and maps 'x, restrict to the zVC of subspace is a v *. x.(g) q-simplices the required simplicial map on of and degen- face The J and B(cuwn, (e), & 9.91, 9.6 and [45, of T(~ Proof. Inview l), 1, eracy maps respected are [45,9.6] operators because the and q are maps e,p, ~O-spaces 1b1.1: are and maps of geometric realizations simplicial of be- %-spaces the zeroth fa~e (obtained from the fl v :zVC v - zV) andofor Ilsrrefore of maps 0 -spaces by map The ym is the limit [45,12.2]. cause 2.4) u -. (1X is a map of ,Q ,,-spaces (by Theorem n:CX the and m the maps wvr: r V C R~ of to by passage induced is QX maps limits from action on of k X)] Cv, jB*(zV, nv] y: I X) CV, zV, B,($ 1 - be verified to easily is and 14.41) and 9.7,12.3, [45, (ace a map of back point is that all requisite compatibility pulls The to the level of finite calculation by !do-spaces explicit (compare [45, 12.41). It would be K compatibility product (for statements inner the from spaces dimensional m to by pointless 1.6, IV. give independently, we know details since the in C) codified and VI. 1.6-1.9 and easily verified that section 2. It is in -spaces. of a map L is Ilir~L 0 I .(g) 5 IV. (a)- in identities specified .I algebraic the satisfy (*) of the maps 3 As we have thus "group completed" the additive promised, Indeed, from obvious inherited are identities these linear for identities iso- rirg multiplicative the along carrying while space olructure of an E identities assertion the given a the by is C% that each and metries w special obvious the Again, behave cases correctly. ructure. by the all of continuity the of holds l(d) I. Condition fig-space. IV.

99 /!lll!l~!~/ll~ll 193 w plication of S2 corollary also morphisms of 8 -spectra. This are be a 8 -spectrum. Then w : TE -+ E Proposition Let E 4.3. 0 rV A. 2(i)] [46, . sefully be combined with the following consequence of w a morphism :Q~TE~ -+ is E is a morphism of h -prespectra, hence fi -spectra. of : 4 projection the Y, o-space ,b a For 4.5. Corollary C'Y -c CY commutative diagrams have we l(e), 1. IV. By Proof. of (k, k)-spaces and, if is an E map ' map induced the operad, .w Q~TG'Y :nmTC~ JI of -c is -spectra fi equivalence. a weak homotopy Ov. B)... v v. v1 l z . A z '(E~A.. CA.*-Au-~vl~. . . ~EV . hEO) Y z E~A.. J~O example, Consider, for the case (e: &) = (Q Q). The I and Proposition oliaries 3. (multiplicative) 3 imply that, for any (WX~ homotopy is QY equivalent as Y, a -space weakly &xY)- 00' 0 as 1) a space. have We to , rDY. When = S Y DY = LI K(Z j' = W and RW, C Vi (j), b 6 role for the of view V.). In B) . . . O g(V1 g completion as equivalent is ,us obtained a group which of K(Z 1) J j' the by played j(g) e .(g) on TEO, it in Eo the definition of, the on to the of I\ Em ring space version as0. This is a'greatly strengthened J follows readily that the diagrams made have we that of the use no Note monoid theorem. Quillen [rrratt- rltructures on and our discussion applies equally well to DSO = #so any 1) for K(Z the of force The I). fi &I, ( operad pair Em j' particular example (dp, Q) is the connection it establishes, via VI. 5.1, thus, sets sphere spectrum and the and finite of category the batween K-groups ?lo 5. 2, between algebraic groups and the stable homotopy VI. first the second part follows This proves the and part, are commutative. Similarly, 01 spheres (both with all internal structure in sight). VI. 5.9 1.6. IV. by corollaries above imply Rand following result. the the 4.6. Corollary Em determined S2w~~@~ spectrum the by ring The 3.3 and 4.3 Propositions imply the following result. equivalent to w (RP~') Q is @Z and, under the category bipermutative Corollary 4.4. a For composite deformation $I0-space the Y, induced ring spectra E of the morphism %SO, r w retraction ,., nWT(crw.) W natural the with coincides Z 6 -. injection split - - R~Y ~O'TQY Q~TCY is a morphism of &- -spectra. is true. The inverse inclusion of R Y in more Indeed, even w 11.101 and [45,9.9 of the deformation obtained from each h and Q~TCY t

100 8 localization and structure E The multiplicative 5. m element -1 prime p which does not divide any of eventually there exists an M fi the previous in as where, 8,g) (X, , (c a Consider )-space ) :H.(Xi; Zp) + ~.(r.(X,0); Z ) n.(p) that such (L sequence increasing J J @* P = is ) K X with pair operad a product &'x ) # , (? ( C' ( section, m' is to when Z 0, = allow p we Here j n.(p). ( all for isomorphism an in P have We a firm on the grasp h -spectrum contractible. and )5'locally rational numbers. 11c the as interpreted m S2 the Clearly spectrum 8) Q~T(X, and its relationship to X. T(X, g ) condition seems always to be practice. satisfied This in is contractible since its zeroth space is a group completion of X weakly invertible. X must delete becomes n mr 0 Thus we in the element which cases. Examples 5.2. X is convergent at M in the following 0 to obtain components spectra from interesting its multi- order in X of grouplike @:Xi X is under 0, so that n 0 X is a ring; here L - Ti(X,B) (I) E structure. plicative a I" homotopy equivalence. weak m - As a commutative semi-ring, We make a simplifying assumption. O-space here result holds by inspection X = CY for the some Y; 2 (It) 4 n a unit of this morphism indeed X; X + e: admits n semi-rings is Z' the of 1551. [26, in ) Z H*(CY; of calculation 0 0 P We assume hence- X. to the unit e: CS' .3 T of obtained by application dividing any the ,not additive translations element of M, p For (lii) 0 an X n + 2' is practice). case in the is (as inclusion forward that e: isomorphisms induce X + ni(p), I j for ) Z Hj(Xitl; Xi + Zp) Hj(Xi; 0 it P 1 a (multiplicative) submonoid of Let M be not M in is 0 such that Z' result since, by holds increasing; here the eventually is {ni(p)] where M at least one element other than 1. Let contains and denote ZM the isomorphism an induces (L@)* (46,3.9], which primes inverting by (obtained M at integers the of localization the lim H*(ri(X, 8); + Z ) Z H*(Xi; l~m j 8); ). Z H*(~~(x, P P P + Define those of union the be components XM to divide of M). elements the for interesting bipermutative = X to example applies last The Ba X of Xm -space h a sub is XM that note and X n C M m that such mr 0 categories a § 5. displayed VI in equivalent is .c) r(X component unit that the prove shall of X. We of M spaces the one application by obtained of considering be shall We operad of component unit the of M at localization the to space infinite loop an as sight spaces all hence id-spaces, in recognition principle Theorem of 3.1 to r(x, 0). X & will as regarded are &-spaces given (where X'-spaces X Fj be - m 3 an group an E space (Y,X) and For element i of the completion m the projection). slmces by pullback along ith denote component ri(Y,y) let Y, n of of the zeroth space the 0 work- by equivalences homotopy invert weak ourselves to allow shall We +r(xM,g) ~(Y,x) of Q~T(Y,x). Let L@:X + r(x,e) and L@:X~ ing Hg category the in II (see $2). 3.1. denote group completions obtained by specialization of Theorem the idea due to Sullivan. the of the following result is In the case X = as0, further simplifying assumption (although it could per- We shall make one [76,5.8] re- weaker somewhat a proved also and case this 'Tomehave proved extra haps be avoided at the price of some work).

101 M, E for m multiplicative means m.1 by translation right thus, case k sult in the X = ~68 I -r(m.) re [77,3.1]. )(x) a group of the definition By b(2). E g fixed any for m) 2(g)(~, 6 = the commutative diagram, following in Consider 5.3. Theorem which composites k-spaces, maps all 'K X ,fl X are 8 of all are spaces - maps 00 of inverses homotopy and 'Xm-maps, kl and the maps i are inclusions of X the is & , where limit taken over for commutative ring any - - components: n m, , HHB(X-:ld) -+ &) H*(Xn; ~(m)*: M. E 6); [46,1.2].) By cofinality, we see'that H*(r1(XM, mpare a) is telescope orphic to H*(TM;&), where xM denotes the mapping 3.91, Moreover, by [46, this isomorphism can be sequence he (*). - the that such Ha in naturally lized by a + XM @: -i map (X Mag) 1 natural (where the is j inclusion): Hb owing commutes in diagram If ) 6 (XM, rl a localization is n X is a ring, then (ri)~~ : X1 -c (i) 0 of M. at X1 with and restrictions cellular under is result the cited ctually, stated l?(rM(X, B), If is convergent, then r~~: r(XM, 6 X ) - 6 ) is a (ii) , weak equivalence. homotopy present to the trivially proof its but generator, one on free transcribes w Z * kl H*(Z X 'Km-maps and element some divides p if 0 = ) X Therefore, convergent, the composite of is if We that first prove ntext.) M' P inverses of kJ X nm-maps This isomorphism on an is multiplication by p will imply that M. that ce an is ) 6 (XM, r1 and -module Z a is Z) H*( ZM; M 0) - rl(XM, 6) B = (rLe)-l(ri)La if M-local will itself XM be least at (or simple is it a localization I? (X, 0) at M. is of 1 0 and 6 from coming H*(X; Z ) on ote the products by P Proof. The last statement will follow from (ii) and from (i) corresponding to the [n] for the homology class r(X, as in M of elements of set the Write 0). order applied to t 0 = ] [p x- that for We claim > q 0. Z (X; H ), P 9 . . . m E M. and cn E $ (n) and define . . m2,. l,ml, n. write = m Fix i 11 deed, by [26 51, ,II. 1. integer any positive for Xn E en(cn)(ln) for n n. the sequence Consider ;53 x(l) * +PI, = x(p) * . . . * = [I]) * . . . x. [p] = x([l] maps and spaces of dm2) dmi) dml the gives @x(P) . . x(l)@. the is coproduct, y iterated ere x Xn - ... -X - Xn - x1 (*I - ' n Xn. - 2 i- 1 1

102 sum in symmetric terms (all p @-factors the same), and y[pl denotes Z if p does not divide any element of M. Therefore the same state- of the P = our claim y q/p, deg Since the %-product. pth for holds rnent y under the of power + rL,:r1(xM,S) r1(rM(x,e),5). divides In many of the (*), p the sequence infinitely follows by iteration. - that implies therefore our claim and H*( z . Z ) = 0. To prove (i), -component 1 the on (ii) M-local, this proves are these spaces and Since m p i M' components. all on therefore X E translation -n choose points right additive be the to p(n) define -n' 4.4 Corollaries of the situations By theorem the of application in p(n)(x) 1.9, VI. and 1.8 VI. definition, that the observe Ad n), )(x, (c 8 = 2 2 5, with an E operad, we obtain the following 4. Recall result. and homotopy is ladder following the that )-space implies of &, ( a m [45,8.14] from by generated monoid commutative free the is CY T that commutative: 0 the based set T Y and that wOQY is the group completion of aOCY. 0 Y, the and correspond- components CLY, Q Y denote the unions of Let C M M QY. C'Y to M in CY, and ing Corollary 5.4. Let Y be a following the consider and -space 93 0 commutative in diagram , X are spaces all xm-spaces in b ~3 which The cited definitions also imply that ~(m) is homotopic to the mth power are Km-maps all and inverses of X b X 'K - )3 induce -r(mi) arrows bottom the Thus maps of composites rknd operation x ' ~~(c~)(x~). m by m. on homotopy multiplication hence the mapping telescope of groups, a ra vertical the Since a localization is the bottom sequence of at M. Xo m Y rl(cY. 8) rl(QY,8) - QIY r1(c1y, 8) py simple KM of a limit (as simple is and equivalences homoto are arrows - - B rl(XM, -. @:XI e) also are (ri)~@ pr pr spaces), XM and L @j = (ri)~ j:X + 1 To prove (ii), note that the first parts of the proof M. localizations at 1 ?t re r J, the commutative X as well as 8) and ladder consider to r(X, apply to rl(cy,t) --r1(c~y,e)~ rl(dMy, o = ~,(Q~Y, r) weak are arrows horizontal All equivalences. homotopy (I) at localizations are arrows vertical All (ii) M. 0 Here QZ = (Q (k', . S = , ) and Y /$I) Consider the case 0 0 infinite loop as space, Thus, SF. = and Q S I) K(Zm, U = an IIMS 1 mtM 5 U K(Zm, 1) ) of our so, (D r I-component lllc group completion is of - I M is Since X convergent at M, the induced map z rM(X, B) of -. meM M a version is This statement M. at to the localization equivalent of of SF mapping on homology with coefficients induces telescopes isomorphisms -. 7 - 5- ---

103 !I It// I! !Ill 1 .. - the paper [76]. of main theorem Tornehave's The force of the particular VIII theor or^* topological and Algebraic . between the it establishes, via VI. 5.1, ($, a) is connection example @ under sets finite of the category theory of the and stable spherical fibrations. apply of here the previous two chapters to We the machinery ring spectra which represent various cohomology theories E btain w analysis The emphasis will be on the construction of and interest. f derived for spaces and spectra from pproximations discrete categories V. J-thbory diagram studied in chapter the to elevant 1, after showing that the In cohomology theories ordinary section spectra, ring commutative th are represented by coefficients in rings E a, category of a permutative or bipermutative K-groups a define higher ring spectrum its associated spectrum or E the homotopy of groups w definition our when a= b$~ for a discrete ring A, ; TB~ Quillen's Ids of A [59,61]. When A is commutative, higher K-groups on construction gives the ring structure trivially K*A. We have ur rather KO*Z in VII. 4.6, and, in Remarks 3.6, we shall lready calculated Beyond these observa- K*Z [60]. results about Quillen's to elate this The calcula- no new ns, d,gebraic K-theory. we have applications to nal power of infinite loop space theory lies primarily in connection with fine structure, such as homology operations (and the arguments in It is not geared 4 structure can be). will how powerful this demonstrate analysis than homotopy types (other owards deloopings of known ones). of of view the present primitive state In calculations in algebraic K-theory, of t is too early to tell how useful the rich extra structure which we shall obtain the representing spectra for the relevant cohomology theories on of the relationship a discussion with end section 1 We be. to out will turn

104 th B Coker that BSF splits as BJ X These J as an infinite loop space at p. between zero on internal structure and the representation theory the P The results [77]. second author the quite proven, first were differently,by spaces bipermutative categories. of spectra derived from richnes.~ proofs the illustrate and lifting Brauer use not present do and In section 2, we prove that the complex (connective) topo- real spaces. Em work the constructions we use of All of structure of rlng E the are logical K-theories by represented ring nmT~@ = kO spectra m 2, but the key calculation fails; orienta- the here equally the at well prime . and kU = nrnT~%. perio- Bott Brauer transport lifting to use We then tion sequence from dicity fZmT~LYk kU to kO and and (q S~~TB & xK odd) all ' q 9 6 9 LB(sF;~~)- '82 J~ SEA BSF away completed the on lifting Brauer that imply results These q. from completed zeroth space level is an infinite loop map, first proven a result may ) be regarded space the zeroth of the 1-component is of j (where J~ €32 2 second author the by [75], case, by in methods. We different the complex J is from B Coker built up space SF as a codification of how the infinite loop Snaith, [8] and of Madsen, Priddy Adams and of also use recent results for model a and discrete J 632 ' author second the and a representation theoretical with together [42], space not of the agree to replace any We CW- a of type homotopy loop infinite an lifting gives Brauer that to prove calculation, a weakly by complex notation of change CW-complex, without equivalent space level. on the multiplicative map loop additive infinite the on as well as the construction weak maps to inverse of without allow to as (so equivalences results is One point of these that they allow us to study infinite loop verbiage). further of the of properties operations, maps derived from them, and Adams by 3. automorphism in section Frobenius the of use 3.2, we obtain In Theorem K-theory 5 Examples; algebraic I. models These V55. spectra j introduced in discrete for models j the P P be a commutative topological semi-ring or, equivalently, A Let result by completion of E ring spectra at p and, at p, the classifying m 6 6 andVII.2.4, ByVII.4.1 an is nrnT~ (seeVI.2.4). .in($L,$)-space B Coker is spherical J endowed B(SF; -oriented for ) j fibrations j space P P , R = x n fact, since E (in spectrum ring and R-spectrum) an infinite with an loop space structure. lifting In 3.4, we use Brauer Theorem m diagram, centering to demonstrate that a large portion of the J-theory an space. ring E -c is I'A space I'A A : L zeroth 4.2, VII. By its m J, infinite is a commutative diagram of loop spaces and around B Coker which com- is A structure a group completion additive the Is of of maps I'A course, Of ; not is a ring. with patible structure. multiplicative its map exponential infinite an construct we 4, section In loop been has A structure weakened to precise structure on original 'I'he #xk roB kr + I' B Bd; higher coherence homotopies. precise to up all only (away from r) and prove that, with r = it r(p), 1 localized at an equivalence an becomes odd prime when Now let discrete, A be discrete. Then rA is homologically domain The p. for models discrete here range and are constructed J and J and the the is nOrA the sense A T = A : L* and 0, > i for 0 = H.rA In that 0 P (29 e' SF map factors through the unit map e: j + I'l B & X kr of . It follows P -- - -

105 lI/1~1~lllll /llll - -. . 205 2 = Ba Then the to n fo~ G(n) roups be BG Define BG(n). 0. is already a ring, then L :A -t rA is completion A to a ring. If of A w p(1): in As explained BG(nt1). 146, BG(n) + the translations of limit is TA R ring spectrum a homotopy equivalence and an thus the E w class 3.91, there is a well-defined natural homotopy ordinary Eilenberg-Mac Lane spectrum HA = X(A, 0). Therefore any ring theory with represented is cohomology a commutative in coefficients t : BG rO~a by an Ew ring spectrum. 5 such that the restriction of to BG(n) is homotopic to the composite associated categories from arise examples trivial Less to com- the ; roBR r - BG(n) -. L r Ba : the translation and Ba of n n mutative rings. from proceed to We the particular. general the : implies that is I'B& -) a group completion L induces Ba fact that a -space, category, then BCC is If (a 83 , 0, c) is a permutative an isomorphism Therefore any coefficients). (with homology on I'BG VII. 3.1 by VI. 4.2, and a map and rBCL gives loop infinite an space X 2. BG to homologically equivalent is : L BCL rC rBa which is a group completion. define the algebraic We To relate the constructions above to Quillen's algebraic K-theory, of by K-groups some e must review of his results and definitions ,621. [59,61 Recall is it its subgroup. equal to commutator if -that be perfect is to a group said a (a ,$, is BR then category, a bipermutative is 1,;) c,@, 0, If sub- normal be N a perfect Let CW-complex be and let a connected X t VII. I'BU by VI.4.4, and (@,a)-space, 2.4 andVII.4.2 give that to X. T of group X up unique , -+ homotopy, frX a map is there Then 1 that (a a is QX~ )-space and X L Xm, with the multi- compatible is and is f -IT f N induces an isomorphism on of kernel the such that 1 additive structure. Moreover. VI. 2.5 gives that plicative well as as the (see with any coefficients homology Wagoner [79]). is If N the com- t associative ring is the graded with a commutative (in sense) and con- then is Y If a simple space. is X a X, 1 r of subgroup rnutator & K *a unit. Additive right translation by one defines a homotopy equivalence no nected subgroup, perfect non-trivial space such contains -rlY that * to the p(l) from zero component I? BU the 1-component BCL . an Y] Y] [X, + is [x', : f then isomorphism. 1 0 have Since we spaces, Em are ) E Ba, (rl and 8) (roBCL, two therefore the record we connection. In this triviality. useful following one 0-connected spectra, coming from the other both 83 , @ from and spaces connected be of a map XI - which Lemma Let f:X 1.1. homotopy as a which have the higher K-groups of of These groups. Then duces an isomorphism on integral homology. * but Theorem spectra 4.1 below will generally be will very different, [x,nz] f :[xt,nz] - interesting certain in do show when equivalent become they cases that space any an 2. Is isomorphism for at an appropriate prime. localized ZX' is an equivalence. Proof. Zf: ZX + be of the form specified in a category let Now the permutative Let unit). The be a discrete ring (associative with A a disjoint union of topological VI. 5.8, SO that a can be thought of as Quillen defined ommutator subgroup EA of GLA is perfect, and

106 t KiA=rri(BGLA) for izl. K,DA - ~,b XA permutative category Consider the the universal .fjXA of 5.2. VI. By in of as thought be can ZA % analo- OA of inclusion the by induced of property the + rOBB$A induces lemma),'T;: (or f a map BGLA gous to complexification. This idea it can be is not new: presumably fi ?' that such PA : r0B BGLA' + Since T 7. to homotopic is f 70 starting the as Hermitian for viewed of treatment Karoubi's point and and BGLA' Since 2/. is so isomorphisms, homology are f b4]. K-theory interesting.) (However, by VII. very KO*Z is not 4.6. Thus are' simple, 7 is therefore a homotopy equivalence. B YA h r and 3.1, 5.8, VI. of consequence VII. following The immediate 0 izl. = K.A K~~XA for It many the topological applications. in role 4. key plays a of 2 VII. to group theory the analysis of the action maps reduces A let Now bipermutative and on is dp~ and 0 h Then commutative. be 6 the Em a bipermutative category from derived FB& space ring we hrA $A thus a ring. Here ~~b K*B~A of is instead If 2. = 0- G(n). Let = use a bipermutative category A of finitely generated projective a commutative graded exists by VI. 3. 5), then we modules (as obtain : and G(pn) 'Tp: - zpJ G(n) - G(pn) xp1G(n) c P ring 851). p. [46, (by 0 2 K.@A i all for KiA = K*$'A such that of be the homomorphisms groups specified by spectra of homotopy as K.A constructions the having Alternative CB . . . CB g -I . . )a(n,. ,n) . g,, = . (g . , gp) (a; c P w-l(l) 0. (PI the groups can be obtained by use of black boxes of and Boardman Vogt, -. n>. c (a;gl, ... and ,..., )w

107 l!l,l tlI/I,,,,! 209 Definition 1.3.. Let p be a let n be the cyclic group prime, contractible be any W let and usual in as embedded p order ol X P' ace on which n acts Em (for example C(p) for operad &). any freely associative) (homotopy H-space X together -space H: n is 8) (X, restriction that of the W X n xP -- X such for each w E W 8: a map with on the product of iterate X. p-fold xP homotopic to the X w is xP to 0 a A and let Let = 64'~ for a commutative topological ring 8') 8) H:-map the such that XI -* f:X H-map an is f: (XI, + (X, An compact group. topological a G be GL(n, - A) For a representation p: G homotopy lollowing diagram commutative: is a subgroup and n of Z define the additive and multiplicative wreath pa product nJp and nsBp representations be the composites to C and nl c --&dGL(n, A) 2 GL(~', A) The proposition reduces analysis of on analysis 8 and gp I'B~~A to P Clearly an an is map, infinite loop an particular in and map, E w products. these wreath of however, there is an essential Pragmatically, 8* I§ [2b, 11 p homology are of terms in defined operations Mod -map. H' m difference. The operation to repre- p in p, hence nJ passes is additive and If X are X'' spaces loop infinite -maps. HP preserved thus are by nnd m to sentation rings, trivially seen and satisfy the character formula is from permutative categories (of the usual form) and if appropriate derived lirnl of determination the reduces 1.2 Proposition vanish, then terms representation f:X theory. -) X' is an HP -map to not H-map an whether or contrast, In m a$@p multiplicative but not additive in is p,'and there is he of details the give remarks following this reduction. no general formula for the calculation of x(.rrJ in p) terms of ~(p). @ Of course, E space are maps an -structure H' induced with infinite loop before passage structure-preserving Remarks 1.4. Let Y be m m only as Y of to choose to we since e letter use the think homotopy, whereas representation theoretic (We techniques apply W xTyP + Y. 6: homotopy. to after passage to following the of This suggests use these remarks of multiplicative, that being appropriate (not quite our applications -1 and Ym components the in points m Fix and m definition. standard) ) section; next the in Y infinite are and on x inverse map Y -1 of Y. Since the product # m maps oop and thus homotopy are diagrams following the -maps, m HP com- 2, the k = p and jr = 'with the diagram of VI. 4.3 implies formula


109 .r'(I"w.sw.rv2vwv.cw follows and @, it @3 that TBG represents the ring-v The external nctor KGX on finite dimensional CW-complexes X. When v W - W X xP and g: W X xP - X ie the X projection on the last is xP defined oneor - KG(X X Y) is KGXWKGY on maps coordinate. (ixgP) 7 = 1. Therefore an H-map X -spaces - Y between HP is m (where the plus notation again denotes rBk g:yt and - rBk - xi H an and only if if -map with commutes it transfer. For examples such as m disjoint a of dition the composite as basepoint) a that diagram chase from BS08 [421. pr: BSO to observe useful simple it is - an E VI. 1.10, of definition, the is 5) (X,e, an that space implies ring E if m m formulatton = of 8 One U. = G or 0 = G for 2 = d or d Let periodicity asserts that ~ott d T 5 and +and . de- where T and and denote the transfers associated to 9 ) QD:KGXWKGS~ - KG(X xs 8 9 note the products induced by and [?,XI. 5 on the functor Lo ah or, equz~alently, that tensoring with a generator lsomorphlsm of author the and Snaith, Madsen, [42] Y.7.2) (below criterion The second an defines isomorphism ] n = BC n r r Bb d d 0 A A A A an H-map f: BU - BU or f: BSO - BSO is an for determining when ,- XG(Z%). - KGX P P P P an f is and if map infinite loop assertion the map translates to that loop infinite ott [Zl] deduced the latter adjoint the verifying that isomorphism by if only it is HP an -map. with The Adams-Priddy theorem V.4.2, together d m no - G composite the of BG useful consequence. following very 11.2.13 and 2.14, yields the --% BG~S~ BG-BG BG 1.6. Theorem of Y be infinite loop spaces and the homotopy type Let X the under the to homotopic 18 Bott map discussed in 15 1. iterated of BSO primes T. of set a at completed or localized BSU of or - H-map an Then - the to L : BG restriction r 0 BG, this adjoint corresponds ~quiv~lencc : d X - Y infinite loop is -map HP an pis at completion its if only and if map an kG -f2 @:kc of the m spaces adjoint of the .emth of 0-components lo for all primes p r T. composite in map 8 H lhb d -k~~f~~l) kG A+ A GAS $2. lifting_ Braucr and periodicity Bott iven n. we in 3.10, - by 11. 2.9. of 12~rB.8 O:~~h f3 that conclude view either for h Write %! or @ categories the bipermutative of = map the with Z BG Bott SB.~ equivalence the under giees X 5.4 and write G for either 0 or U. Define specified in VI. X Sld(BG - Z X G Z). a3 11.3.14, immediately that kO By 11.3. 2,II. 3.9, follows kG = f2 TBM . and it Bid homology isomorphism The T. - BG : ; is a 0 in spectra connective to the obtained ring isomorphic are kU cnd ~.d equivalence, homotopy hence the zeroth space TB~ is kG of to BGX equivalent Z. Since from the periodic Bott spectra by killing their homotopy groups in nega- map a is TBG - BG(n) .Lt : L of H-spaces We have 'thus proven the following result. ve degrees.

110 ( ((1 : ( ' ( ,( ( (;(< (I ( ' '. , i: : :: ( (. I:' ' $: (, (!!(..'(. , 215 214 kO and kU represent real and complex con- following diagrams are commutative: Theorem 2.1. ring-valued nective cohomology K-theory (as theories). 22L RkH X \(HXG) RkG now In particular, the diagrams 1.2 Proposition of reduce the AXA[ analysis BO and BU to representation theory. of homology operations on 1. additive diagram to BO and BU was first justified Application X RG of the RH . x G) @ R(H [unpublished] and been exploited by Priddy [54] in by Boardman has diagram also [1, 4.1(vi)], the commutes, following By formula Adams1 [lo.] K-theory. in Snaith and by mod 2 homology where, for r qa, dr denotes the iterated Frobenius automorphism = 2.2. Remark by Bott As proven real factors as the [21], periodicity of the two natural isomorphisms composite RkG * 4 4 ZS~(X~X) and KOX @ %sps4 - @'%sps4 -., KO@ Y). I A RG transformations these context would seem of full understanding our A in a theory of to require E module spectra over E ring spectra. m 'Thus ~(p) = A(p) if p: G -c GL(n, k) factors thzough GL(~, m ). k We turn to Brauer lifting. Fix a prime q and let k = now 'The real of the diagrams above also commute in the analogs case and 9 k denote algebraic closure be the field of q elements. Let an the of to ROG. ROkG relate the orthogonal representation ring r field with r = qa elements contained in k, so that k = lim k Fix an Of passage to classifying maps and then to Grothendieck course, - r' * Recall - cC of multiplicative groups. k* from Green embedding p : ring homomorphisms RG groups gives and - KU(BG) KO(BG), - ROG 11 if Theorem that [31, p: GL(n, G is a representation of a finite - k) and become isomorphisms when the left sides are completed. with these p(g) has roots 5 .(g), then the complex-valued function group if G and Moreover, by [14,4.2 and 7.1 [14]. the respect to IG-adic topology n 0 (and dimensional a finite is p. 13,17)], KU-'(BG) = KO-'(BG) and 22 x,(P) i(g) ye = i= l the represent BU -c kr) BGL(~; r): (n, A Z Let vector space over (virtual) a unique of character the is representation E RG. 2.' to of lication the if Quillen [58, p. 791 proved that k), O(n, in takes values p odd is q and GL(n, k) and the trivial x then is representation the character of a (necessarily unique) real P and additive, G in natural is A Since n. degree representation of anotherfinite H+GL(m,k) -n: If ROG. E h(p) arepresentationof is increase. r and n h(n, r) are compatible (up to homotn py) as maps the then H, group unique hornotopy relevant vanish, terms %mi Since the there result = x,(~) .I. xp(g) x,~~(~* g) x,(h)xp(g). g) ~~~(h~ and = Therefore, if RkG the the representation ring of G over k, denotes

111 , l,,i, , I I i 217 if q is odd, A : BO -c BO (*) : A BGLE -c BU and, inclusions terms vanish, t after further composition with the lirnl q 9 Now conclusion is immediate from the the BGL(n, X kr) BGL(m, ks). compatible with the A(n, r). Quillen's the of proof The step in main [58,1.6]. following conjecture was Adams result the result: following implies the (B) Similarly, diagram of (*) maps A The 2.3. on induce isomorphisms Theorem 2.5. Lemma commutative: diagrams homotopy The following are q. Z for each prime p # in cohomology with coefficients P r0sllPk ad ro~Bk r0B@k * 6r * '" in section have explained As we of the group completion 1, property the recognition principle 1 gives VII. 3. homology isomorphisms $r 11 I. P : BU - 0 BO B - : BGLk GB hxk 6k and 7: -c L roB B0k be convenient to it will this At point, (and generic introduce a we define Lemma 1.1, Invoking be used throughout the rest of cation of notation, to ,u if ro~hXf; BO X : -c BU and, + q is odd, A : r0BKq (**) 9 book . for shall Y' for specified "discrete write models" We .- - such that A classes 0 homotopy to be the unique we course, (Of X. C? case, In each y6 will Y. ologically significant spaces or spectra the invoke its but construction, plus use could also the properties of of discrete categories by means derived from the classifying spaces would discussion.) the to nothing add have follow- In particular, we the topological suitable constructions. observation. the following need shall We 6 2.4. homotopy commutative: Lemma are diagrams following The and 2.6. completions be the to BO BU~ Define Definition y from spaces (with and Xi; q . q of the b roB odd) 9 9 and roBk;fk~~o~~X'k Lr ro~~~knro~b~k~r0B~~k, 0 Bbyk 0 = G that convention the to ert U = G or and define 4 h q ). to be the completion away from B~[l/d (** of the map A of G~ + @ 4 ABU BU XBU - BU E ne ko6 and ku6 to be the completions away from q of the 0 w is BG~ hen O-com- the QW~B TBBE b~f; Q spectra and rOa r Bb dPk . BO and k and replaced by BU similarly and with 9 9' 0 6 . kG nt of the zeroth space of for any %U(XAY) -c ~U(X X Y) is Since a monomorphism Proof. of 2.3 consequence immediate Theorem result is following The suffices it Y, X and products smash with diagram the second consider to characterization the homological (see completions of $2). T[ suffices to prove diagrams, it both For replaced by Cartesian products. A A a homotopy is B~[l/d + :BG& A equivalence. Corollary 2.7. and thus, since rele- the 'T; X with after composition commutativity

112 This justifies our of BG' as a model for BG. Of course, thinking of rather than localizations is essential here since completions the use can BGLE' 5851. p. now = z[~-?z [59, We n 0 = nZiBG~~+ and Zi+l 9 q 6 verify that represents the kG away from q of real or completion categories bipermutative from passage by induced is fir Proof. A - infinite X the equivalence that K-theory and connective complex an is spectra m to . ring Em to ring E to completions away from q spaces map. loop two the implies that 2.5 Lemma hY in same map induce the composites 6A 2.8. Theorem There is isomorphism a unique kG kG[l/d -c &: and spaces zeroth on of ring spaces. The one conclusion this map that is of ring spectra in HA such of A that the 0-component of the zeroth map and 11.3.15 11.3.14. the of clause uniqueness follows by A is equivalent to 0 G = U. or : BG' - BG[l/q], G = 6 A 6 Let the of 1-component the denote kG of @ zeroth space BG6 . iso- Proof. no(kGo) and nO(kG[l/dO) are both canonically L Clearly ~(l, 9 ) = Z2 . GL(I ,* 9 ) = 9 * , and the from away completion morphic to the ring ;[l/dj = 2 X and there (continuous) a unique is ' (P) P# 9 q the infinite loop map B p : BX* 9 -) BC* is an equivalence by a simple of isomorphism of fr,om one to the other. rings this isomorphism Denote 6 Let cover connected simply the denote BSG* homological calculation. n no? and By a trivial ?. . 2.4, Lemma from diagram chase 11.3.10 by 0 A following as of BG6 @' The same observation. yields the 3.1 V. of that proof together an BG[l/d 6 BG6 : equivalence determine and A 6A 6 6 : the composite b6 Write X for kGO - kG[l/d0 of r'ing spaces. infinite equivalent are BU* and as BO@ loop Lemma 2.10. A to @ BSO~ X BO(1) spaces to and 8. BSU~ X ~~(1)[1/61 6 6 the zeroth map of BGg of - 1 The -component BG : i3 kG6 -ndkG Thus [b6] [b] . Let = hh* adjoint be to the composite A 6 6 with compatible clearly is kG[l/d by the given the splittings A: kG It is by the map infinite an loop therefore following 3.1. V. and lemma 6 6 and , zeroth spaces satisfy the of hX the map P ), p), (&[l/d, (kG Then theorem. those A from conclusion follows 11.3.14 (and 11.3.9). The hypotheses of A: is -+ @ BSG an infinite BSG[l/d loop map, Theorem 2.11. 2. results and 11.3. G=U. or G=O this result is following of topo- addendum The is the reason that that prove suffices to the completion By Theorem 1.6, it Proof. logical automorphisms Frobenius that the shows interest; it fir and prime q so does if it hold will this and HL-map, an is # p each at A of Adams operations (both completed away Jlr q) agree under A. from notation, of clarity For G. by replaced on the localized level with SG H&, case real the in needed point additional The only The commutes in diagram following Theorem 2.9. U. = G case the treat we G=O G=U: or s that the relevant representation theoretical constructs, in particular

113 I I I I I I It 111, 22 1 220 homomorphism, the andthe requi- restrict decomposition appropriately, r; argument, to prove this with suffices an it limit obvious replaced 9 contained in the appendix of &illen1s paper site information is [58]. each r = qa) in the top row. (for k 1.1 and a transfer argument, Lemma By M Let with so if it domain commute homotopy will diagram resulting tile does the the be let p, to prime integers positive of monoid B = X qF q' M (since kr) GL(~, of H subgroup a p-Sylow for x~(BH)~ restricted to W L let be Em map, and Y * :X natural let ), 4, F(x, = let Y the a is ). Z = ~(B,#xc, 5 Z since and p [he index of T~H in .rrJGL(m, k ) is prime to used interchangeably be can U. and course, Of BXG me bxC given here, its usual topology. By VII. 5.3 (and the diagram p-local space). Let P be a finite field between k and . By a being 9 in its have infinite loop maps statement), we chase, suffices to prove that the diagram above homo- diagram it -trivial 4XHq,e) + Y and $:rM(B k

114 lifting honenst than virtual representation is crucial. an to rather By (applied to Proposition 1.2 fact $ under that f the C a), an is infinite are spectra and spaces all this section, Throughout com- to be prove diagram it suffices to chase, that the com- a trivial map,and loop r = 3 Thus r r(p). be to is odd) (q qa = fixed pleted and prime at a p is BGL(m. BU a) X {m) -!h homotopic to fl and that BH~ posite a generator and p2 mod reduces r of 2 = p ~f of units to of group the the outer of the diagram is homotopy commutative. rectangle Actually, retain We 2. > p if Z con- and previous section the of the notations p2 * * kq because of 0; - the homomorphism p: lift- in our definition Brauer of with the discrete models for various of the spaces llnue of discussion off in Adams certain by operations. general be will assertions these ing, $3. J-theory in the maps .~nd V diagram of need we rectify this, To only choose with the consistently p requisite have We an equivalence of orientation sequences a manner do We (in homomorphism decomposition p). of independent * BSF ~BO~ LB(SF;~O ) SF follows. this as Let Z = in (q) = 620 B0 Inductively, Z(d. = A. let and I 7 1: I II 1 B A and B~-~. let j-l, localization be of 2 of the ring Aj-l, the given A. J BSF - -B(SF:kO) ABO SF 8 integers - 1)- ~[exp(2.rri/(qj-l)(~'-~ = B cyclotomic chosen (q-I)] at a j of equivalence an and 2.4) V. (compare f fibration sequences prime ideal contains which a Let the field be a K. C C Bj-l. J j 6 cur) BSpin 6 - B(sF;~o~) - BC' of ideal, let be 1. quotient the by its maximal A of fractions of A let €3- spin@ j J j' be I + A r: let the + A i: map. be quotient the inclusion, and j j j - k Moreover, A contains a group Obviously char l = q and lim I = spin@- - BC j s' j j - P v roots onto of maps isomorphically (qJ-I). . . (q-~)~~ r of unity which j 6 spin6 and . . BSO BSpinO liere is the 2-connected cover of * ls ltS €3 8 the corresponding subgroup com- being these isomorphisms , I of V j j cannibalistic cur) is the (defined above class universal space. loop We specify p:~* as patible j varies. to i? by 5* its restriction letting j q ffr by determined 2.2) V. because map loop infinite an is and ko6 -c ko6 : - 1 r we diagram, last In agree to be io our . the construction of choose ring spectra. of BC6 fibre The of E fir the completion of a map is m P A = A. and 1 = l. for j sufficiently large. charac- It is then obvious that the J J endowed an of it as BC c(br) is thus think infinite loop space, and we P of ter so is a + H i5: x =+ is that 5) ~oB(i$) GL(m. p. Similarly,if G P' Theorem in We shall prove 3.4 space structure. an with infinite loop order greater than pe and if finite p-group with no elements of infinite loop of diagrams commutative are above diagrams both that A) u: G + GL(~', is a representation, then the character of maps "paces and is 0 6 and xr, is C) GL(~~, + G iw: x{mp) BG therefore BU B(iu): + of the infinite loop space In order to obtain a better understanding 8 GL(~', A), it follows that With u = ? 0 (11~): H + p . homotopic to P BC: , we construct discrete models for the spectra j and j02 of V. 5.16. P of rectangle outer the commutative. indeed is homotopy our diagram w VII.4.1. the functor $2 T from Em Recall to E w ring spectra of ring spaces

115 6 Definition 3.1. Define j jQ2 SlrnTBnk = = and Slw~~~k3. 3 2 For p > The bipermutative categories 2, j = SlWTBb~kr. define P the VI. 5.7,5.3, and 5.2 (and in are bxkr specified nk3, @kg, and understood to be completed at p). E ring spectra are Let specified w 6 th of the denote and J J the 0-component and 1-component zero @P P A 6 of space and let J02 and j J6 is X Z equivalent to (which P P (PI)* 6 6 and 1 -component of (jO ) denote the 0-component 2 0' SOQg I passage by induced are c0 labelled arrows The Since 2, p bspin. tx bo > results of The following theorem is based on ideas and Quillen bipermutative of Kq ;f h spectra the inclusion from completed to Priddy [28]. [57, 591 and and Findorowicz 5.11 all are arrows dotted the 2.8, Theorem and [l, By ategories. 6 and 3.2. j : are equivalences v Theorem There -jP P On the level of bipermutative categories, d . H in quivalences 6 - the : commute in diagrams following HS that such j02 A j02 v: Xkr. By passage to &XK ax%- restricts on kj to the identity q 6 6 i6Kk~6 and j,-; the composite completed the with $r-l of spectra, that conclude we 1 is trivial. map qt of inclusion the by induced ku6 - j: $$ rkr in q .-- 6 br F and p iso- obviously induces an , 4 results a lift There j p: P 1 - the K6 bipermutative categories where induced by inclusions of are group G [14,4.2] = 0 for a finite BG morphism on To. Since KU 6 6 by induced ko6 + kU is p (when that j sense the 2, > in functor zeroth space since and the the [48,~111], fibres with commutes P determined zeroth map of p is the by the hnmotopy 0-component of It will be convenient to treat the cases p Proof. > 2 = 2 and p commutative diagram - separately. discrete obvious the adopt We notations the of analog models /' in V. 5.14. B A BU~ u6 (fir-1) fibre the we have Theorem of In view 2.9 (and [48, I(2.1 2)]) , (i) p > 2. fibration sequences in following comparisons of where and , H& F(jr Quillen [59, p. p proved that pO induces an isomorphism on mod 5761 Ffir denote relevant fibres: the homology andis therefore a homotopy equivalence (since J and the P It follows that p of ibre fir-1 are p-complete simple spaces). nduces isomorphisms on T. for all i and is thus an equivalence. I 1 - c +jpis .A-p. V: j: desired he equivalence

116 3 3 3 6 (ii) p = 2. Let and the fibres of -1: ko6 -+ bso FO$ F$ denote 6 3 and of $ -1: ko6 + bspin . 5.15 and Theorems 2.8 and 2.9, By V. fibrations of acommutative diagram comparisons yield 3 in the maps which A are equivalences. bso6 -r $ -1: ko6 of composite The 6 K~: j~2 -r ko6 is trivial since b3: @K3 - eK3 restricts to the with identity on kg and since [j0: , bso6] I [jo6, bo6] by the proof of 0 2 0 6 0 5. 15 and the £act that H V. = H kO j~; (where H denotes mod 2 3 F$ a manner that such in 8 and induces right triangle commutes The 7 and FO$~, + j06 )I: obviously results cohomology). There a lift 2 is arrow diagram solid 8 and the the fibre of is canonically equivalent to induces on an isomorphism rr0. 0-component to of Restriction the restricts non-trivially Slbso6, to 9 a braid fibrations of I [48, (by (2.13)]. diagram zeroth spaces gives a homotopy commutative construct to 5 p. Re- define = 9F. We need a slight calculation and we 2 6 3 2 that call kO Since H [8]). by g., (e. ) (A/AS~ Z1 ~*bso 0, = * 6 3 we have and an exact -. kO induces the trivial on H -1: $ bso6 map 8 equence 1 -.. A/ASq + ASq2 0 . 0 - EI*FW3 -c Z(A/AS$) -c not is ii[0 and Fiedorowicz but the diagram, by Priddy determined Here 13 have [28] to the restricts 8 class non-zero unique its Z2 = Fo$ H and homotopy makes the triangle which & any H-map that proven Thus * 6 and is therefore commute induces on mod 2 homology an isomorphism a By inspection of the fibration H of generator Slbso . [29], ideas Quillen's up following homotopy equivalence. (Friedlander we fibre, * 0) with K(P(2). j02- FTF+ is this that see 0-connected 1 16 particular equivalence cohomology earlier [57], dtale obtained a about the (e. fact known consistent with is BOk3 H = J02 H that g. [28]) - equivalence an is necessarily Thus map.) pO , not an infinite loop the determinant, to the with non-zero classes corresponding the consider Next, H in & : diagram following in H & . the norm, their product; spinor denote and last of these classes we the fibre equivalent to In view of VI. 5.3 and VI. 5.7, J: is 5,. by 6 6 6 is j02 + jZ of cofibre the Thus of [28]). 1) K(Z2, + J02 SO: (see the cofibre by the long exact homotopy sequence. Clearly, )

117 6 6 map X(Z j02 be the non-trivial map e , hence j + $1) must 2 2 (by (2.12)] [48,1 conclude We e. rrlust equivalent fibre the be of to 6 exists there that j2 - F d3 which p the diagram above commute : makes The desired p is an equivalence by the five , HB and lemma. in - 6 - 6 p. and A p A. j v: equivalences and j02 are jZ * v j02 * : it follows = j6 + ka6, K': from the definition of K~ Since P 6 6 that the kO - BSpin@ (in V § cur): ) restriction of c(Br) to B(SF: 2) There results a lift is the trivial infinite loop map. B(SF; j 6, P 6 6 the proof of and 5.17 yields the following V. j ) -) Bc6 5 : B(SF; P P ' corollary. is c6: B(SF; j6) - Bc6 infinite an equivalence of 3.3. Corollary P P loop spaces. was regarded as a ring spectrum by pullback along v-' In V. 5, j P - to v restrict composite On of zeroth spaces, v and l-components equivalences - 6 6 where $'/I: Fob6 the and of denote BOB- Bsph6 and F$@ fibres @ Q3 3 theorem that the following the We shall see in $ BSO~ /1: Bo6 .4. of '8' e9 maps in view of Theorem 2.11. maps A may be regarded as infinite loop not 2, = even p when easily seen to be an H-map; is 2, > p When PO the However, clear is this much the of FoFiO. of view in non-uniqueness = 2) following proof will yield a possibly different (when p of the theorem 6 map P@ J8 * F $& which is an equivalence of infinite loop spaces, Foa3 F J06 + of argument gives an equivalence and an analogous ~b: @2 Q3 composite The in ApB plays a central role the infinite loop spaces. "multiplicative Brauer lift diagram" displayed on the following page.

118 The dotted arrow the diagram is an elaboration of part of the portion of = V p, with r at r(p)). The follow- 8 3 (completed of J-theory diagram that result yield an approximation to this part ing discrete asserts models by spaces diagram of infinite loop diagram of the J-theory a commutative all that this fact consistent with in pre- and maps and approximation is space structures in sight. In assigned geometric words, infinite loop other discrete notation behaves as if it were a functor naturally hoc models our ad the identity. equivalent to Theorem arrow (-) and dotted arrow (--3) The 3.4. solid Brauer lift diagram are braids of fibra- the multiplicative portions of horizontal ( t-, ) arrows tions, all equivalences, and the the are diagram a commutative diagram of infinite loop spaces and is entire fibre j B(SF: j P ') is equivalent to the J&- of q: B(SF; T: P ') - BSF. maps. loop infinite 5 and ~:B(sF;~o')-BSF induces <:FBIC'+J @P (bybasechange). By [48,f.(Z. for 13)], (which is a precise form fibrations), of axiom Verdier's the Proof. the of portion arrow First focus attention on solid - 6 that such is there equivalence a canonical FBK' ~:FK' -+ and diagram. features two j ' It kO ) and orientation sequences (for P and {OL~LOQT err. the comparison between them. obvious We must construct an infinite such that : t' 5 ' induces an equivelence Clearly, FBK' + Spin loop map 6. 6-p -6 -6 6 .spinQp ti= LOG nc(plr). r = 5'0~ and and K6 such J' - 2 F TF: fibre the to equivalent is I,' that of to - J' @P ' It remains construct Define 6 ' = q(~')-~:~~in& QPe 6 6 equivalent to the is ') j B(SF; - Spin* 76': j ) - r6 FB a fibre B(SF; P J;, - F $&, pp: we note that P and with infinite loop these equivalences being compatible of B K6, 6 = "i' -Spin O' ngr/l: r;. ~c(g~)anr Z6oL0n7 (i60g)ot. *' Qd equivalences @: ') B(SF; r6: j p and - BC ' - F$&. ~hus con- J' P P The constructions so the all result by passage to zeroth spaces from far and L letters the $3, V as in (in which, diagram follow+g the sider n nnalogous constructions on spectra, hence we may regard the diagram natural of the for generically used are fibration sequences): maps notations By abuse, we retain the . ns of connective spectra in H one Here cofibrations and fibrations level. the spectrum the of diagram on / --

119 ,j, I , I 8 i 1, I I I' 232 233 6 [48,X1], standard arguments hence with by sign, to up agree cofibrations such kO) B(SFr -.. ) kO B(SF; B'lk that is an infinite loop t there map 6 induces T6o F @& such that 6 ti that show = and BSF -- ) q:B(SF;kO = q~B'A roA *B(SF;kO) B'A~T:BO~ 63' p@ P 6 5 and p@T = 6 05 J6 in , HA We BA, infinite loop maps. homotopic to must verify that B'A is = th TO qo BAr A r BAo T on the space level. Thus q and course of back passage and to zero Now p@ five the by equivalence an is lemma. 6 as factors /B'A some map w : B(SF; kO ) +B% . for and rw the Brauer spaces and comparison of multiplicative diagram above to the 6 BC type X BSpin the has ) kO B(SF; Since . * T of homotopy of latter portion lift diagram the complete the proof that the solid arrow P' w maybe [BCp,BC@ V.4.8, V.4.7and y = 0, by V.7.8, since and diagram is and a maps spaces and infinite loop of braid a commutative as a map BSpin egarded the BO@ . It clearly induces - trivial homo- have already level, space the On we constructed the of all of fibrations. cohomology, orphism on rational (indeed, on integral) and V. 2.8 and horizontal equivalences, infinite loop course assign of could we and space may BIA. now r specify We null homotopic. Thus it that imply 10 BA is to structures requiring them ranges by their to be infinite loop maps. Similarly, map o r, as an infinite loop (BA)-'. by c(+~) = Ae c(gr) new problem that The the consistency of check is resulting to remains the specify Bv by j ) be to B(SF: requiring as space loop an infinite infinite loop the geometrically constructed infinite with structures space P 6 -1 infinite loop BA~BK we map = BK specify and : + B(SF; (Bv) jp) BOW BSpin@, B(SF; kO), and structures loop already existing on space (SF; kO) and r =. Bv 0 T -(&d-': j J@ - B(SF; loop infinite as ) their spaces. loop Lemma and 2.10, 2.11, we V.3.1, By have that Theorem P P trivial. remaining verifications The are maps. 6 + and A: ~Spinl - BSpin Boa Boa A: '8 63 emphasis followiilg part of the theorem for single out the We may therefore specify +'/ 1 as an are infinite loop maps. both We (compare V. 5.13). infinite loop map by +I/l = An gr/l0 A-' (compare V. 7.6 ). The 6% rA J Corollary SF composite The 3.5. @P bp F@@- fibrations F$' - J obtained by comparison of A: equivalence QPP cLa as be taken may -c J of the 3-theory the :SF & map diagram. P @P given an infinite loop space is infinite loop an then J if map is QDP At odd this mainly 2. the prime of force at The assertion lies E specify Next, =A%xe:SF-J structure as the fibre of +l/l. P @P almost trivial, since have we there it is primes On the space level, parts of the multiplicative as infinite an map. loop 0 imply that then to commute Brauer lift diagram already known [SF, [BZ~, SO SO = 0, ] r [Q~S .sod s '1 QD '8 that fact is uniquely determined by the and xe:SF r E 10 ~S2c(+'): J L -. that so [14], and VLI.3.4 by ~~0C2qr (SF;kO) p P €3~ SF xe: to homotopic is BO -c BO T: J -c its composite with 8 V in 3. € map labelled the for conditions were defining the These 63- QD 63'~ P the digress to give We of application following the corollary, as of view in BOB maps infinite loop Xe Axe:SF = = that have We -rr& P hich about [60] K*Z. summarizes Q uillen' s results (by once, conclude Delooping we [48, I(Z.12)]) 7.9. V.

120 Therefore, 3.5 V. and Corollary by 4.6, 2 if by multiplication odd. is i Remarks 3.6. a have we For A, ring any commutative topological of is a summand direct K *Z and the image in K*Z of the element bipermutative of diagram commutative categories zJ8i-1 Quillen [60] proved that order of 2 in in 2J8i-5 maps to K*k3. xero Z by that Adams' e-invariant noting maps J K monomorphically to 4i-1 2 4i-1 the T 4i-1 X induced by unique lift the identified with map can be -r4i-1SF X of the Pontryagin character to the fibre -, BO SF - X of 5: Xe: SF (23 commutative degrees a derived diagram of K-groups in positive and that observing and 5 necessarily factors through 4i) K(Q, X B0~4 ir* Chern of representations because the discrete classes of groups B/Y~z 1 r a direct sum- not is 2J3 [34] found that Karoubi torsion classes. are Lee deep Z Z K that result , proved the [37] Sczcarba and and K of mand 3 3 is exactly ' Z 48 translation from by first the from results diagram second where the to 1-components of where 0-components and zeroth spaces of spectra odd at SF of primes The splitting 54. and 3.4,4.4, VII. via identified with is SF 4.6, VII. By 4.5. rlB& e the of summand a direct onto monomorphically maps T: = K KO+Z, spaces and to Again, all spectra a fixed at completed be are * s m a. IR, A complementary summand = isomorphic to T*(RP When being ). r(p). Actually, almost all spaces and r = q 1s to be q f prime p KO*A = K*A = T*BO that the element k. e shows 5.6 and V. T.SF, i 5 1 tn groups, hence localization will homotopy sight will have finite 11 @ Z2 8, summand defines in K.Z. mod 2 or a direct the J. denote Let with completion. agree P' in = of p-torsion k the image 2 and A > When p T~SF. j*:~.Spin + sequence orientation Theorem 3.4 focuses attention on the r(p)' to the p-torsion subgroup of K*A is isomorphic and, by J* Corollary P L A @P s6 B(SF; SF LBSF. jp ti ) K*Z. J* is a direct summand of 3.5 and V. 4.6, Finally, consider P zJi' map T The of is homotopic by the splitting null SF in V. 4.6-4.8 K.Z in 0 is unknown, i 5 or 8). 2 i (and 8 mod 1 image of The Z = J. 21 2 show = [70,9.11 or that 26 11.12.21 2, (and Corollary 3.5). p When of 2-torsion The subgroup KO k3. = A Let k3 Kqi-*'n = k is 2J4i-1. 4i-1 3 splitting is SF there no J the = C2 X and 2 as H-spaces, presumably 6 JU2 and JU2 the U13-~: BU - BU and for rOs /i d~ k3 Write fibre for of null be to delooping hrst fails already T of 2. > p When homotopic. at (completed 2). there 2, 3. Theorem of equivalence proof an By the is h. split to it use shall and Jkr shall for we an exponential law prove - 6 6 natural the which under JU2 + JU2 vo: map JO * XU2 corresponds to the 6 that SF B(SF; kO ) as infinite loop spaces; it will follow and T is map [1,5.2], by complexification. By induced JU2 + SO2 c: -r4i-1J0 and 2 infinite loop map. an as trivial same are c group and the is the identity is and if i even -r4i-1JU 2 4i-1 -

121 1 I I 237 n and 1 are to the 0 the restrictions and SF J6 -c e: M denote the Let e: Q SO& J will M monoid {r Subscripts n 10) aps . ] 0 P @P Q,S' free the is M. Since components indexed on denote unions of of course, components of the zeroth map of the unit j6 . Of the P 0 the Qx~ also and 11.1.6) (by - free spectrum generated by S we but be to p require odd, r6 and J~ equalities which involve @P P is an exponential spectrum generated by SO (by IV. 2.4 and 2.5), there diagram analogous a precisely construct can specified on by S 0 6) , 0 firn~(aMs + 0 6 QmS er: of map unit spectra e a 2 and 1 map * 1 -c x for any chosen point x E Q So and also a unit 0 0 co ej: A* s[ T(B of X -spectra e: QmS *fi Q h;C'kr, 8). By VI. 5.2 and 5.6, unit e: have a functor we functor forgetful and a bJ'kr + 6 J2 J@2 ecy !& fe: = er Let . $ + kr X f: of (which is an exponential map 2 6 + b + from = ef: Recall qkr b$kr. let and categories) permutative g than ~f & xkr. An analogous diagram can also use nk3 rather by 0 VI. 5.1 that B 4.4,4.5, VII. and freeness notations, these With . DS = $ that p assume we Henceforward, 0 kr. constructed be use of by the following homotopy commutative diagram in which 5.3, yield and 5.4 is odd. all by indicated maps are + J6 : a6 equivalences and " homotopy SF VII By the results of dia- our maps in all 5 cited and above, $4 P P to defined to corner: the composite from the lower left is the upper right be of gram are composites of maps homotopy and Km-spaces X X X 6;1 X ,-spaces, diagram the and a: k induces a inverses ' of maps of are Clearly all three operads similar commutative in diagram HA. 0 required: it is naturally which acts on Q Urn 0 S , Y on SF, and B on of sources geometric different k X kr (in two ways). Because the our is the actions, the statement that all of of maps preserve them highly non-trivial. 6 a ; is J6 SF A J The composite 4.1. Theorem P @P a homotopy equivalence. our diagram, also be By specified composite may the Proof. - described as 6 6 8) J = & p- 5) rl(~M~lkr. a 8) $kr, r0(eh & = rl(B.ulkr, P equivalences L localizations are they since (ri) are $ maps the and Here -1 on isomorphism induces an .I'g It clearly suffices to prove that €3 further localizing or p. at M and we are completing all spaces at The

122 239 238 elements there 81, [59,5 Quillen By H*. mod p homology exist y. of k to Qs[l]*[-p] QS[l]+[l-p], degree and of 2i(p-1) 2i(p-1)-1, zi degree 21, such that i translate the decomposable under elements modulo of the *-product 5 6 By the multiplication from the zero component to the one component. kp) P{ H*BGL(m, . = E{zi} (E3 yi} H*J P H*SF immediately that the [2&,11.5.6], follows on it # for table we shall only need that H Actually, J than larger no additively is P * composite 1 - This part of Quillen's work and depends only on the is easy stated. (er)* (ri"")* %SF I QS[l]*[-p]} {p E 10 QS[l]*[-p] P{ HASO C form of the p-Sylow the the of subgroups kr) GL(n; 573-5741. p. [59, of The rest H of the computation the argument to Ji will fall out of monomorphism is a 2). = (this the step which would fail if p being follow (and Brauer is thus lifting). of independent The remainder of Since through e: Q )-'e SO - J6 (rio~ we conclude by factors ' P o r €3 general on the of proof depends solely the Pontryagin pro- properties a count of dimensions that ducts operations QS and 6' determined * # and the homology and b *[-PI [-PI} ~~[1]* 1 } P E{ Qs[l1 p = H*J: respectively 9 and 6 [26, I1 $l,2], together with ring on by spaces E m a Hopf algebra (under A. over the Steenrod algebra Moreover, as *) 0 of particular properties Write juxta- [26;154,11$53. H+QS # by by x translation a basis have now in we x*[l], * know we which class homology the for a com- of [n] position on elements, and write 6 (because A over a coalgebra as J6 and and 0 the are @P P J6 H.JB n. ponent By II.Z.8],modulo linear [26, of *-products combinations already kr). that know We components of r~h 1 between positive degree elements, 1 k Q1[1l* [l -PI # 0 (Q 111 *[-PI) $:(rg)* = S 1 -' E p] [rP- * r)QS[l] (rP- - . Q [r] (a) P 1 - an isomorphism. is j% techniques that standard by follows it and (Tg) * * p2 to a generator coefficient is non-zero because r reduces mod The 1 - of In algebras detail, Hopf connected of a morphism is (rg)* $* the 0 group the of or e x [26,11.2.8],for By H*QmS Z of units of . P in an isomorphism same finite dimension will be and degree each if it is x c H*T,(B bzk,, e), of degree 2s(p-l), ps, Let primitive elements. monomorphism on a mx (x*[l E m])[m]. - (b) namely the degree be H of basic primitive element J6 even th s the * P' r): imply = r-pb(rP - k then (a) and (b) Let [26 by Since, 1.11, ,I. sth Newton polynomial in the ~~[1]*[-~]. S '- s(p-l) (-i)r(r, a p,fQS[l] calculation gives a standard - Pr) ~'-~[l], p])[rP]. - *[1 k(QS[l] E [r] Q (4 0 Ps P*~ - l)ps-, - Pr - s(p-1) (-llr(r, = Since e , exponential, are rg and .Z) they r(QMS - US' : send S p when 0 2 k s for pk = unless 0, > r 0, # Therefore some P: some [T-~]. [r] Q to [-p] + ~"[l] k22 if 1 follows $-IFg td-le (ria that and it (c), of In view send

123 11'1) ,\,I,; '11 ' I 241 map The 4.3. Corollary and P ps = 1. prP-l P~+~-~ = k and pi pstp-l = 2psml if X (cD~)' LO 6, j B(SF: ) BO + W' B(SF~ P QD 1. 2 Let bs. Thus, by induction on s, $;'(~~)*(p,) # 0 for all s equivalence infinite loop s an of spaces. of primitive element degree basic odd sth the be 1, - 2s(p-1) degree j6) through factors BSF Since q: B(SF; -. Proof. P elements decompos- modulo , so that bs - p QS[1] *[-p] 6 H of J , q: from this B(SF; kO ) - BSF, follows V. 4.4 (last line), V. 4.8 (i), P 4. 1 sf again Since, by [26,I. 1.11, * under able sbs). = Bps (and (-1) the $3 V together with the multiplicative of J-theory diagram and - s(p-1) (-l)r(r, = pQs[1] P: gives another calculation l)i3~~-~[1], - pr of Theorem diagram Brauer lift 3.4. 6 6 ) (SF;j Cp -- of original diagram The this section suggests that P de- analog of the additive infinite loop space C6 multiplicative the case, hence in that the even degree The coefficient here is the same as @P and J the as of e: QOSO -+ J' . fibre ed that lulow we course, Of - 1 P cases the same special show that (rg)*(b P since s all for 0 f ) $* equivalent infinite although In C6 spaces. loop contrast, are The proof is complete. Q'P P t-'(rg)* (bl) # 0. of c6 d is homotopy evidently are there equivalent, equivalence no P # for the product In the following corollaries, we write or * infinite loop spaces between them because their homology operations according on infinite loop them of think to choose we whether to spaces differ [26;I $61. I1 54, 3.3 that Corollary from Recall multiplicative. additive as or composite The 4.4. Corollary (SF; as an infinite loop space to is 6, j LIB(SF; equivalent = ji) P 6 C' . OBC = P P (but equivalence is a homotopy not infinite loop map), m6 T-' is an Corollary 4.2. The composite.^ P 6 XOq ff x*l. where T: Q So -C SF is the translation x 6 0 * SF #--) SF x SF (SF; x J j 6, P P proof 4.1, Theorem of our and 41 [26,I§ the image In by fact, and -la' the algebra over an as H*Q~SO generates Dyer- )* 0 ( infinite of equivalences are spaces. loop re- and as), this statement even Lashof algebra (under * and the hence the equivalent to e: SF fibre of J~ E. LIq is - By have the also we II$l], [26, following 2. mains true at the prime @P ' the also the theorem implies that the first composite, and thus second, the in been used has technical consequence our of homolo- proof which on homotopy groups. induces an isomorphism BF and gical study of here have BTop we Conceptually, [26,11]. in crude information used geometric the to obtain operations homology on map w: BSF loop Choose an infinite -+ B(SF; j ') such that P as infinite loop maps. wq = 1

124 splitting we the geometric splitting to ob- there and SF, of used tain more subtle information. 6 4.5. Corollary takes : elements H*SF + .J~ H ) the (n P* *P of the subalgebra to generators i3QS[1] and [-p] QS[l]* *[-p] and 1 com- is homotopy commutative, where the maps v are the 0 '3 Theorem of jp - 6 th algebra considered as under H*SF the 5 product. an of ponents of the zero the of map equivalence v : J I 6 -1 higher operations The point is that no con- 1, > l(1) [1], Q vea v and Thus ecr6 is an equivalence. E 3.2. n P P P P 6 )* image of tribute (a to the the specified on generators. Since - SF is an infinite loop map while a . J -" SF makes v-': n J P P P P' P 6 with is multiplicative respect to on HGF, , rather than 2, # (%)* and, is such the particular, J-theory diagram commutative homotopy in 6 such operations can contribute of image (a ). on decomposable to the the 5 and In view of Corollary 3. j: Spin that it. through factors SF + P * elements. to maps are two it is that hope natural these homo- previous remarks, as be chosen that a can least, This at or, topic map. loop infinite an Remarks 4.6. The Theorem of proof original gave 4.1 author's second P (the the on conjecture Adams complex and think may we odd, J of information. Since different is p certainly V.7.14 if hold would P were level) infinite loop satisfied. as the fibres J of. BU and BU on Jtr/l and Jtr-1 . Since QD @Q 1 reprint] recently, Friedlander [Stable Adams' conjecture. Very [J U] 0 = kr) BGL(m, KU- = , (by 3; 2 Theorem [14]), the com- and P Unfortunately, have announced V. Conjecture of proofs 7.14. Seymour and a posite equivalence & is characterized homotopy by of 4.6 V. com- P P machinery Segal's proof, hence Friedlander's to seems essential [68] of the mutativity diagram structure loop infNite SF on ours with agrees yet known it is not that his such basic (and is ours essential to facets of the theory the orientation as We argument. Seymour's of details the seen not have sequences). the pr is determined by the cannibalistic +r and standard class where orientation BU B(U; -c calculation representation theoretical A kU). the ([77,4.1]) shows that diagram

125 I I , I I , i ', 1 ;I,~II I I I 245 -c loop space theory to : KA A KB infinite K(ABZ B). Suppose given a functor E from rings in Pairings IX. written EA = {E.A I i 2 0), such that E A = KA and suppose -prespectra, 0 t by natural E understand a we which admits an external product, tensor iring EB) (EA, zeroth of p map the extends which B) @i E(A + spaces. D r E -c F are central to of spectra of the general form Maps is [27] has proven that a natural map there th Fiedorowicz data, these a The this of is to develop chapter purpose stable homotopy theory. such that fo: E A = KA GWA -0 KA is the identity. -c of R-prespectra EA in guise the to recognize such maps of allows one pairings that theory 0 follows connective associated the that 2.11) and 11.2.10 (by spectra EA of of appropriate space level maps are X,Y, and XhY where + Z Z, equivalent. d GWA are an Em space, such a theory X.AY will not itself be Since Em spaces. m denote let EA Now S2 TB PA regarded as an R-prespectrum. is the recognition already principle in not implicit certainly obtained rtainly homotopy type). to (up KA is EOA The of this chapter results prespectra with (weak) work to convenient be will It VII. chapter in imply that E 11 admits an external tensor product and thus that EA is and their pairings in the sense of Whitehead this [80] throughout e associated GWA. of connective spectrum stable cate- and notions these between relationship The the chapter. related monoidal symmetric of of pairings develop suitably We notions I1 53. in has gory been explained and ategories, of permutative of categories, We section 1. in 62.-spaces obvious While pairings is an of desideratum of any com- a theory 2. section in Q-prespectra of pairings induce %t.-spaces of rove that pairings infinite loop of theory treatment plete and should have many other space summarized follows: be chernatically, our results can as attempts for the need applications, to compare our it emerged in m monoidal symmetric Pairings of categories $'A finitely of category the TBpA,where is R machine-built spectra up blow a ring over modules projective the Gersten-Wagoner to A, generated U pairings permutative of categories and [30,79]. Let column be the ring of infinite, but row CA spectra of by CA finite, matrices with entries in A and let SA be the quotient C of pairings 'Q-spaces Wagoner and Gersten by the finite matrices. the ideal generated + that QKSA is equivalent to KA, where KA denotes BGL(m, A) X KOA. showed II RwT i pairings of $2-prespectra = {KS I i 2 0). and thus produced A 52-prespectrum GWA an BGL(w. A) among projective modules, cofinal are modules free Since the While relationship evident intuitive between an is there present regarded be as may limBAut P, P compare PA (up to homotopy type; E * E and the eory I have chapters, the of ring theory not attempted earlier 03 the universal property of the plus construction (above VUI.l.l), By 851). [46, p. more a single into thought of lines the two combine theory. general induces a map QZB) (A FA PB + X functor the product tensor

126 1. categories of ?&spaces of Pairings and categories be products symmetric monoidal with and 8 a, Let , diagram commutes for all permutations cr E 2. and T E 2 and the following pairing @ that such €D and units 0. A @ : a functor is X a 3 k' = with together 0 0 bidistributivity a coherent natural B @J unlabelled isomorphisms are iso- given by the the commutativity where dg = A 0 0 and of morphisms : and , 63 , isomorphism d: (A e A')@(B o B) B')) (A'@ B) B') ((A'@ @ B')) (A@ 61 B) ((A@ s (*) the since needed are parentheses extra for B, and o& E A' A, E B' 663 ; not assumed to be associative. 8 on is really The category theorist will a definition. not is this recognize that of the By of commutativity isomorphism use a determines (**) , would require elucidation of the meaning of coherence, via a Precision (*). diagrams commutative The in natural bidistributivity isomorphism d as d unity, the and associativity, of which diagrams specification involving just above give coherence. Thus a pairing of permutative categories is also a and c are required to commute. and commutativity isomorphisms a, b, pairing categories. of monoidal symmetric We be to prefer to exercise It instructive is an in those The details La Plaza would be analogous to [35]. that verify MZ'A X HZB -C B~(A BZ B) course, a pairing of permutative @: Of the example to is be clear. the informal since intuition should categories. objects 8: PAXW-, il](A@ZB). sets single the have inmindis of generating the case, this In keep element 1, (**) is between relationship the diagrammatic and only trivial, a pairing requiring by categories permutative of to define like would We the isomorphisms need be checked. commutativity isomorphisms the right of expansion However, maps. identity be to the d monoidal Q of VI. 3.2 from symmetric categories Recall the functor (*) when the sides.are left sides of ((A61A1)@AS1)@(BCeB() and (ACB(A'~JA~~))@(B@B~) permutative to categories. demonstrates that, in the absence this requirement commutativity, strict of Proposition 1.2. A pairing 8: monoidal X 8-r of symmetric CL led to the following definition. would be unreasonable. We are naturally determines X a pairing @: 1Pf3 - categories per- Q C: of Definition 1 83 , Let .l. with categories, permutative be and , such that the mutative diagram categories subsets .ba of Da and products @ and units 0, and assume given *.c~r @ Qaxmn generate ..I X a 8: 063 which pairing and Cn under @. A of 1. 7,- x.2 = all for and 0, = 0@B 0, such A@O that 8 a functor is sequences *c @ &xJJ , A.) of objects in B(X and all {A1,. {B , B 1 of objects sequences . . 3 k isomorphism. natural commutes up to coherent in ha

127 >!/I ii,it~i 249 Proof. (Pa is the free monoid with unit 0 The objects of space of and hence e'4 a take , in we as the generating Oa set by generated yxxy\ z the product on again 8, Recall that, with as written pra for similarly 8 . n on is by objects specified ,Ak (jk) ?Q X .- 2jk . .8 A.) = Al 8 (A2@ (A3 8.. . (Aj-l =(A1@. A.).. .)) . 8 J J [46,4.2] or VI. 3.5, the morphisms from-A to A' in QJ-. are the in As Yk y1 X." ..., Yk' Xlb ) . Y1' = yk) "', Y1* XjA *xjs ... (xl, ..., ".., J in , with composition and the commutativity & morphisms from to ~TA' nA c data on these from way evident an in determined isomorphism @ . not have a definition (or I do any prospective applications) for a notion A -c A' f: morphisms On by objects. on @; + @'a Qax @: Define (**) for F -spaces course, a general E operad & . One of could, of pairing m A' @B A @g: f morphism the @B' Qd% and QCL in B' + , B g: and imply appeal be replaced can -spaces & that shows (iii), which 2.7 VI. to tlie is composite specified by Pj G in &!-spaces. equivalent y Proposition If @: 1.4. (1. X iB .-, F is a pairing of permutative then f ategories, B = X @: Ba B'G -L. B& -spaces. is a pairing of Q determined & , where the unlabelled isomorphisms the by are uniquely in (regarded Ba is given by the object 0 The Proof. of basepoint isomorphisms a, c, isomorphisms coherent and d of and are the natural Bn zero. Ba nullity the by of A through factors and f s a 0-simplex), The commuta- for required the diagram in the statement of the proposition. of The the diagram of Definition 1.1 implies the commutativity following in tivity of the diagram Definition the the from follows .I 1 of coherence diagram: coherence definition X @: . Indeed, the omitted formal 83 of -, pairing given coherence here specified simply by listing diagrams which those be can the proof. suffice for present of The tensor pro- operad E ~154. Now recall the categorical m Notations functor induces a 1.4) (of VI. duct Z. + Z.XZk J Jk functor space the classifying of and thus, by application B, -.. zk X 5. J jk Q(jk). --, Q(k) x (j) kj @: a map -, Let Definition X, Z Y, 1.3. and Z be -spaces. A pairing f:X X Y the previous in v map was the as just defined is v ere functor the through following is and Y Xh such that the factors f which a map is diagram of B. efinition. The conclusion follows upon application commutes:

128 The pairings recognition principle for 2. $3 to a VII recognition principle of one extend operad here the We pairings. Although the recognition principle is basically for present theory elaboration the additive theory, it will of be necessary, for techni- an still X is constructed from 1L Qm(j) XX~ by Prodf. use of appropri- D m convex bodies reasons, to work with the little cal -fi. operads n rl unit and p product its and identifications, basepoint and equivariance ate n. Defir: cubes operads little rather the than with maps the from induced are structural 1 unit and y and Q operads the for Xm(j) )Cm+,(jk) + )Cn(k) x by 63: principle in The proof consists of a check, and [45, 2.41) jjm (see . 1.2 VI. cc c x cl; ,..., c. xc; ,..., c. xc' > . x c1 ,..., c.>dD

129 /It/ i I I Ill i I I I I /Ill I ' 1 1 252 253 ,m+n is given on little convex bodies c by conjugation ,G homotopy e required .m+n 'mn. '7.1 .m~mxA xnDny zmin Y) A (DmX~DnY) Dm+&X h with the orthogonal transformations ul(c) t ' pnI 1 L+n pmA not just the existence but the form of the homotopies that is It is lh7hl A z~'~(x * y ) Z~X AZ~Y ssential for purposes. our result is following example, the For immediate Proof. We may define -, Km+n(X~Y) just as in the X-: KmXI\KnY the previous proof. tfrDm - Moreover, as was krnn(~fi"r). ah and mn = then previous proposition, Proposition part of the following diagram commutes 2.4. The bottom out in pointed in the case of little cubes, the following diagram is [45,8.3] all the top up to homotopy for commutes and part Y: X and commutative: XhDnflY / 'm, n+~ D m adjoints. to passage by The conclusion follows morphism of operads specified h u -r : n+ is the Recall that a pairing. At this point, [80] we recall Whitehead's definition of for and also write u little on 1 X convex bodies, c -. c by T -+ A pairing $: (TI, T") 2.5. Definition of prespectra consists of inde- We need to know that, up to homotopy, v is : 1 5;1 X u :+ -c n+l. '- n aps Tl * Id-: T T;A n m, for the to up such that, 0 2 homotopy, mf n The following of analog pendent of the choice of privileged coordinate. following part part commutes and the top diagram of the of the ttom dia- + ;. be specified u': rin Let little cubes will the idea. give [45,4.9] for .L~+~ sign (-1)": the to up commutes by -C 1 X c on little convex bodies. c maps The u and u1 from *kn(j) to ';~:~+~(j) are 2.3. Lema homotopic. Z;-equivariantly J by Proof. Define orthogonal transformations g, gl: R~+' Itnf1 - - if is even n (s, x) (x, = and g(s, x) = gl(s, x) s) odd (1-s,x) if n is i n n x for there E s and R e g' a both have degree (-1) , Since g and is R. n n For a ,@ specified X, we have the prespectrum TX -space , we R -+ have convex c: R body a little For I g'. from O(n) -9 g to path h: g'u'(c)(g')-l = ul(c). gcr'(c)g-l = u(c) and

130 u hl T maps = X, with structural X where (4 (4 0 mq ynsn Dn x*srnt,s1A om X*Smi1*(4ynSn 1 mf ad4 if 1 .. a(a = B(I.U,I):ZT~X = + ,p.x) . B(~~~~,JJ~~~.x)=T~.~x 'mil, n, q 1A-T 1 \ 1 for pairings. the recognition principle following We have 1 (4 (4) n $ \ umin,q_D(q) mnq ,,,(4 ZnSminAS1 ZnSminil D YnS AS XAS~AD~ m min minil Theorem 2.6. A -spaces naturally of pairing f:X&Y -c Z b /' of prespectra such that $ TZ f. 8: (TX, TY) + = induces a pairing 0,o the maps inthe first diagram Proof. of As 2.1, Proposition - can be iterated to yield bottom first The part commutes by Proposition 2.4. Provided that twist we to 5" of the homotopy of Proposition 2.4 application D(%nD('Y past S1, (4 m n m = Sn A (wpich S spheres smashing with f, Smin composing with By D milin m in give which orthogonal the of and that homotopy to S transformations spaces and here), are taken as one-point compactifications of Euclidean top q homotopies These yields a homotopy for the diagram. part of the as map using a twist 7, maps obtain we with are compatible varies each the face and degeneracy operators (for i( so on passage E I) and to determine the required homotopy t parameter the commutative By the definition of a pairing of -spaces, we have for face with compatibility this operators is It geometric realization. diagrams Dminf 'mn rather convex bodies that than little cubes is essential. use of little D XhDnY- m Dmin(X"Y) - w from pairings of prespectra to in maps via passage the While $2 been we and category II. discussed in 3.4, already has the stable 3.3 more elementary passage from pairings say perhaps the about a bit should on projections and the given actions of are D composites 9 where the of willing Provided that we are of of SZ-prespactra. prespectra to pairings and X, Y, Z. of and face the of definition the operators In view degeneracy can be redefined homotopically SZw functor to neglect phantom maps, the of commutative and 2.1 Propositions 2.2, diagrams it the and [45,9.6] (nw~). , d~~~~ Tal = simplicial a map is Bmn* of any For simplicia1 spaces. follows that given r : T f(S2w~)i being with by the oth term of the system. limit spaces based homeomorphism V, and the natural U a pairing Given maps the T, -+ T") (TI, $: U induces a map IuI A Iv] -.L I *V I, and therefore we X V] U 1 Iv] Iu] X $ obtain a map to geometric realiza- TmXATnY TminZ on 8 passage + mn On Certainly of diagram = f. tion. the level of q-simplices. the $ 0,o the previous definition can be written as follows:

131 Bibliography .. 'ij "i+j+m+n nm+n+l ;I nm+n n T +n- Ti+jtm+n si"~;+~^ Q of Annals spheres. on fields Vector Math. 75(1962), J. F. Adams. 603-632 J.F. Adams. -I. On the groups J(X) Topology 2(1963), 181-195. 11. groups J(X) - the On Topology 3 (1965), 137-171. Adams. J.F. J(X) On the groups 193-222. - III. Topology 3(1965), J.F. Adams. in the upper part which can- there being a permutation of loop coordinates J(X) J.F. 21 Adams. On the groups 5(1966), - IV. Topology -71. Still neglecting the definition in inserted the sign cels of a pairing. Adams. Lectures F. on generalized homology. Lecture Notes J. maps, maps there result phantom 1969. Vo1.99, Springer, 1-138. Mathematics, in OD - ) nnT:' ~el(G?~T~+~t_n = (Q~T')~A(S~~T"). 87.: T)i+j (a J+n J 9 Adams. homotopy and generalized homology. The J. F. Stable .- diagrams The following nmT. - (SlmT',CZmT") $: a pairing give which of Chicago Press, 1974. University are clearly homotopy commutative: Uniqueness S. Preprint. of BSO. Pr.iddy. J.F. Adams and Anderson, W. D. 1963. Thesis. Berkeley, Anderson. generalized D. W. Simplicial K-theory and homology theories. Preprint. are There W. Anderson. in D. phantom cohomology operations no Preprint. K-theory. of context the In the of Theorem 2.5, the group completion property Hodgkin. D. W. Anderson and L. Eilenberg-MacLane The K-theory of the by fiOO is characterized recognition principle implies that the map 317-329. 7(1968), Topology complexes. = i = j 0 of this diagram (compare [46,3.9], VII. 1.1, and the para- case and Topology Clifford modules. A-Shapiro. Bott, R. Atiyah, F. M. result). above the latter graphs -4 Supp.1 3-38. (1964), 3 phantom One of , without pairing a genuine $ neglect could obtain B.Segal. Atiyah and G. M.F. completion. Equivariant K-theory and just the discussion of maps, an given by elaboration mapping terms of the in Geometry 1-18. 3(1969), J.Diff. However, the extra precision [43, Theorem 41. cylinder of techniques M.F. Atiyah and D. 0. Tall. Group representations, A-rings view in be insignificant would 3.4. of II. and the J-homomorphism. 8 (1969), 253-297. Topology

132 259 t J. P. and Barratt G. M. Eccles. E.Fsiedlander. Computations of K-theories of finite fields. Topology f I, 11. - -structures Topology 25-45 and 113-126. 13(1974), 87-109. 15(1976), On the Amer. Bull. K-theory. algebraic of spectrum S. Gersten. Becker H. Gottlieb. and The D. and map C. J. transfer fiber 78(1972), 216-219. Soc. Math. 14(1975), 1-1 4. bundles. Topology finite characters linear of groups. The the general J. Green. J. M. Boardman. Stable homotopy theory. Mimeographed notes. A. everything J. and R. M. Vogt. Homotopy Boardman H-spaces. M. Math.Soc. 80(1955). 402-447. Trans. Amer. Lissage des (1968), 1117-11 22. immersions - 11. Preprint. A.Haefliger. Soc. 74 Math. Amer. Bull. The K-theory of some more well-known L. and J. M. Boardman Vogt. Homotopy R. Snaith. and M. Hodgkin P. V. invariant algebraic Lecture structures on in topological spaces. Notes Preprint. spaces. / Mathematics, 1973. Springer, 347. Vol. M. Karoubi. Lecture Notes ~eriodicit: de la K-thkorie Hermitienne. les th&orkmes de p&riodicite/. remarques Cuelques sur in Mathematics, R. 301-411. Spr-r, 1972. Bott. Vo1.343, Bull.Soc. Math. France 87 (1959), 293-310. Mathe- for Coherence distributivity. Laplaza. Notes in Lecture M. A note on the Amer. Bull. bundles. sphere of KO-theory Bott. R. 1972. Springer, matics, Vol. 281. 29-65. Math. Soc. 395-400. 68 (1962), ~oincar; Lashof. Amer. Trans. cobordism. duality and Math. R. K. A. Bousfield and completions. limits, M. D. Homotopy Kan. K. 257-277. SOC. 109(1963), Vol. Mathematics, 304 Notes Lecture in localizations. and Szczarba. On H. K R. (Z). Preprint. R. Lee and 3 1972. Springer, maps from SF to BO at the prime 2. J. Ligaard. Infinite loop H. Q9 G. Brumfiel. homotopy groups of BPL and PL/o. On the Annals Preprint. 291-311. 88(1968), Math. operations Eilenberg- the in Homology Ligaard Madsen. J. I. and H. H. al. Cartan et ~hriodicit; des stables d'homotopie des groupes 143(1975), 45-54. Zeitschrift Math. spectral sequence. Moore groupes Bott. dtapr:s classiques, ~Gminaire Cartan, Henri 1971. Mathematician. Springer, Working the for Categories Lane. Mac S. 1959/60. the On Dyer-Lashof the algebra of action H*(G). in Madsen. I. and May. J. P. F. Cohen, T. Lada, Iterated of Homology The Loop 235-275. 60(1975), Math. J. Pacific in Vol. 533. Mathematics, Spaces. Lecture Notes Springer, 1976. in maps Infinite loop V. Madsen, and Snaith, Tornehave. J. P. I. Z.Fiedorowicz. spectra A note algebraic of on the K-theory. geometric Preprint. topology. Preprint. Lecture of spaces. Categories and infinite loop spectra J. P. May. Fiedorowicz Priddy. S. and 2. finite orthogonal and spaces Loop 1969. Springer, 448-479. Vol. 99, in Notes Mathematics, groups. Bull. Math. Amer. 81(1975), Soc. 700-702.

133 \ of intern. Congres Actes, groups. Math. Cohomology Quillen. D. 57. May. operations on infinite loop spaces. Proc.Symp. J. P. Homology Nice 1970. Tome 2, 47-51. 171 -1 22, Vol. Math. Pure 1971. Soc. Math. Amer. 86. Quillen. The Adams conjecture. Topology lO(1971). 67-80. 58. D. in Notes Lecture The spaces. loop iterated of geometry May. J. P. K-theory Quillen. On the cohomology and of the general linear D. 59. 271. Vol. 1972. Springer, Mathematics, field. over groups a finite Annals Math. 552-586. 96(1972), J. P. May. E spaces, group completions, and permutative categories. a3 D. 60. dated to July 26, 1972. Letter Milnor, Quillen. J. -94. London Math. Soc. Lecture Note Series 11, 1974, 61 Higher K-theory for categories with exact sequences. Quillen. 61. D. J. Memoirs Amer. Math. May. spaces and Classifying fibrations. P. Note 95-103. 1974, 11, Series Lecture Soc. Math. London No. Soc. 155, 1975. D. 62. Higher Quillen. K-theory algebraic I. Lecture Notes in The Topology. Algebraic of Homotopical Foundations J. May. P. 1973. Springer, 85-147. Vol. 341, Mathematics Math. Monograph London preparation. In Academic Soc. Press. Amer. Bull. Quinn. Surgery on and normal spaces. F. ~oincari 63. Math.Soc. J. May. Infinite loop space P. Bull. Amer. theory. Math. 262-268. Soc. 78(1972), To appear. Quinn. 11, and Geometric bordism bundle theories. F. 64. surgery and Milnor J. algebras. Hopf of structure the On Moore. C. J. preparation. In 1-264. 965), 21 81 (1 Annals Math. symplectic J-homomorphism. The Ray. N. 65. Invent. Math. 12(1971), to Introduction 0. T. 1963. Springer, Forms. Quadratic O'Meara. 237-248. Patterson and R. R. Duke Math. E. R. J. Stong. of Orientability bundles. 66. Bordism Math. J. Illinois J-homomorphisms. Ray. N. 18(1974), 39(1972), 619-622. 290-309. of BSO P. F. Peterson. Mod p homotopy type and F/PL. Soc. Bol. spaces. G. 67. spaces Configuration Segal. and Invent. iterated loop 22-27. Mat. Mexicana i 4(1969), 213-221. 21(1973), Math. S. the classifying spaces of Dyer-Lashof operations Priddy. for theories. cohomology 68. and Segal. G. Topology Categories 13(1974), certain matrix groups. (3) Oxford &art. J. Math. 36(1975), 293-312. 179-193. , 69. J. P. Serre. ~e~rLsentations lineaires des groupes finis. Hermann Annalen 169(1967), D. Puppe. Stabile Homotopietheorie I. Math. 1967. 243-274. part I. 70. V.P. Snaith. Dyer-Lashof operations in K-theory, stable homotopy category. Topology and its D. Puppe. On the 103-294. Lecture Notes in Mathematics. Vol. 496, Springer, Budva 1972 (Beograd applications. 200-212. 1973), 1975.

134 - I:<( (.('( ((! (i (. ('()(.( ( (.(, (! ((I( (( ( (,I( (( (( (( \ ( (\(I(! (i (. 3 I 262 8 1 3 1 D. topology seminar. 71. Index Mimeographed notes. Sullivan. Geometric I; il Princeton. 1,; 5 I topology, Part I. Localization, periodicity, 72. D. Sullivan. Geometric 74, 34 category monoid 80 Boardman stable Abelian Galois symmetry. and T. I. M. Notes. Mimeographed Bottmaps 16ff,213 ad am sop era ti on^$^ 106,Illff, 217, Genetics homotopy theory and the Adams D.Sullivan. 115, 132, 215, of 219 73. conjecture. 213ff 17, Bott periodicity Math. 100(1974), 1-79. Adams conjecture 108, 136, 243 Annals 214ff lifting Brauer multiplicative 220ff, 228ff R. Preprint. L. Taylor. Observations on orientability. 74. 132 I 120ff, bo, bso, bspin 76 I Brown-Peterson spectrum T. Tornehave. the Quillen map. Thesis. M. I. Delooping 1971. J. 75. bs; 120 bu, 76. Tornehave. On BSG and the symmetric groups. Preprint. B_X 186 C-space 9, 142, 188ff J. 146 free I primes. Tornehave. The fibration theory at odd spherical B(G;EJ 80 splitting of J. 77. (C,G)-space 194ff 145, 188ff, 55ff free 146 B(GV;E) homotopy groups of 149 Preprint. 122, 228, 235, 240ff unit of 147 B(SF;j) A. Tsuchiya. classes for PL micro bundles. 78. Characteristic 112ff, 116, 135, (C,G) [Tel 146 B(SF;~O) 123, Math. J. 241ff 43(1971), 169-198. Nagoya 9 CCTl B(To~/O) 129 monoid in 19 algebraic K-theory. B. Wagoner. Delooping J. spaces in classifyhg 79. 112, 132, 223, 228 BCp 114 C Topology 349-370. 11(1972), Cp 112, 132, 240ff 106 BC~ W. Whitehead. Generalized homology theories. Trans. Amer. 80. G. 241 BF, 99, 240 BSF 93, P c: Math. Soc. 103(1962), 227-283. 103, 123, 219 cr 106 Boa BSO, self maps of 101-102, 109 C(X) 119ff, 129ff infinite loop of maps 212 C-algebra 146 130ff, 141, structures on loop infinite Cannibalistic classes 96ff 111 universal 96, 106, 120, K-theory of BSpin, 223 102-103 108ff, lllff, pr Adams-Bott 106 242 BSpin@ 116, 132, BTop BSTop, Sullivan or 124, 132 123, 127, 138 BU@ 103, 220 bar construction 54, 182ff Category 9 18ff, P7hitney on 20 bipermutative 127ff, 154ff, sum 159ff ~arratt-Puppe sequence 33 Permutative 152ff, 158ff, 204ff, 246ff Barratt-Quillen theorem 169, symmetrical bimonoidal 153ff 152ff, rnonoidal symmetrical 199 1931 186, 246ff Block bundle 129 translation 157

135 li,l!)\ classical 23 15ff, group LA 162, 164, 224, 235ff 21, space 72, 140, 146 Em ring 21, 23 19, space cl-assifying 140 Em space infinite 93 map loop a category of 158 95 equivalence in C[Tl 140 66, GETe] monoid 188 a of cobordism groups 76, 84 G(n) G/PL , 129 /PL(n) G,G(n) 22 52 cohomology 39, E-orientation 52ff 53 canonical GV-bundle 50 orienta- operations in cohomology G 92ff, theory of a bundle 52, 56 E-oriented tion theory 82 61-62, 95ff stable equivalence 91 of 93ff perfect of hombtopy groups 8 1 1 cohomology theory 41 206, spectrum Gersten-Wagoner 6t equivariant half-smash product tTr, 4 108 J 114 244-245 Coker 147 units exponential J02 228 223, 122, 215 149, Grothendieck group 97, diagram 100, 135, comparison 137 245 product tensor external group completion 150, 168, 178 37ff, 116 completion 17 grouplike sequences 171 composable f 209f ~E-s~ace 14 product composition M 195 34 HS convergent at (X) , JSpin eP (X) 11 9ff ;I JSpin 54 FR, FE, SFE cross-section 50 HT 34 space 15, 19, 23 homogeneous J-h~momorphism 91 217 (superscript) 6 bordism 78 27 category homotopy 6-invariant 11.8 J-theory diagram 107 34 groups homotopy k 159ff, 248ff, D-space 150-151, &I 253ff sets homotopy 55 33, 12 fibred products -space 160ff 150, (D,D) smash fibrewise product 51 3 9 in 53, theory orientation (D,D) 150, 158 0, 82 60-61, I[Tl 12 in monoid 20 d-invariant 1 18 164, +r Frobenius automorphism 223ff 219, 215, 217, homomorphism 221 decomposition spaces 26 12, function determinant 55 bundle KO(X) 115ff 32 function spectra sums direct 27 f 1 1 Sf KSpin (X) 217 discrete model K,A 206 rx, r(x,O) 183 11 3-functor 234-236 K*Z group-like 20, 50 €-invariant 119 66ff G-prespectrum 50 20, group-valued 207 KO,A ' 75 236 161, 164, sets) E(=finite 50, 20, monoid-valued -space 66, 140 G 234 K0,Z e-invariant 120 66, G-space with zero 72 177 KvX. operad Em 140 G-spectrum 68

136 monoid of reduction structural 141, 145, 177, 72, Monad 179 K-theory 50, 77 functor a 181 action on 201, algebraic 204ff, 234-235 action space 141 a on 214, 218 connective 214 ring representation 141, 146 algebra 213 43, 121, periodic 181 177, partial space ring 45 237 165, 224, Nk3 N-space 148 22 Pin 16, 148 (N,M) -space 203 148, (N,N)-space N 126 P 129-130 23ff, SPL PL, ITr 124 S (=Sphere 34, 7 1 spectrum) of pairing 85 77, space normal monads 250 246ff permutative categories 248ff 0-spaces symmetric categories monoidal - 246ff weak 42, prespectra 69-70, 253ff 37ff, 115 Localization 127 22, 23, STop B(SG;E) 99 of of k0[1/21-orientation 123 periodic space 43 (X,O) TI 196ff of ring 45 space "spectrum") also (see SF 199 of to obstruction 8lff E-orientability 15, 22 plus 205 construction Sp loops kO-orientability 118 6 26 nV 240 jp) R-orientability (SF; 57 76 Pontryagin-Thom construction "spectra") see nm (for reduction Top 128 to df E- 187 spaces S-orientability 59 27 prespectrum of spectre 34 coordinatized 3 1 141 operad 139, free 29 on a space 140, 142 action 41 R on an operad 143 action suspension X- 29, 70 semi-ring 148, 203 10 linear isometries 68 pseudo 148 E ring little convex bodies wgak 40 171ff, 176ff 169, 69 product 35, smash 32, M(G;E) a,4ff 173 169, cubes little Puppe 34 theorem desuspension partial 181 172, 176, spectrum 28 75ff MG M(G;Y), 0 28 pair 144 operad 36, connective associated 112 M 145 action on a space 47 44, P 29, 71 Slm partial 178 associated connective 35 108 2 52ff . n-connected 35 orientation ree 33 canonical 58 coordinate-f MG-orientation 77 coordinatized 31 68 ring E orientation diagram 76 MSPL MPL, 80ff, 204 36, Eylenberg-~ac~ane 83 30 free 83 MSp-orientability periodic 43ff 223, 81, sequence orientation 44ff connective periodic 235 110, 223 r(p) MU-orientability 83 periodic connective ring 46 46 periodic ring recognition 188, principle 183, 35 ring 191, 254

137 23 Top/PL spectrum unital 66 22, Top 23 weak ring 40 Top bundle 127ff spherical fibration 50, 128 117ff, 127 kO-oriented transfer 211 23 Spin 16, 22, 104ff of kO-orientation trivialization 59, 77 59 stable spinC 22, 23 16, orientation of 104ff kU 164 norm spinor 163 U G-bundle 91 stable SU U, 22 15, 17, 95ff 92, E-oriented 16 bundle universal Stiefel-Whitney class 57 E-theory 81 f f universal 1 2 covering spaces 1 18 k0-theory suspension zv 26 (for zrn see "prespectrum") 34 of spectra 54 W w(S;E) 81ff I W,W 110-111 WHS 41 95 weak equivalence 34, 51 11, sum Whitney 15, wreath 165, product 206ff 110 classes Wu. 51 49, complex Thom 110 52, isomorphism Thom spectrum 65 Thom zeroth 29 space of a G-spectrum 179 192 and Q~T of 186, T

Related documents