When are two commutative C*-algebras stably homotopy equivalent? Marius Dadarlat and James McClure * Department of Mathematics, Purdue University, West Lafayette IN 47907 1 Introduction Let X and Y be finite connected CW complexes with base points, and let K denote* * the C*- algebra of compact operators on a separable infinite dimensional complex Hilber* *t space. The purpose of this paper is to study the question of when C0(X) K is homotopy equ* *ivalent to C0(Y ) K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued* * functions which vanish at the base point. Recall that two C?-algebras A and B are said to* * be homotopy equivalent, written A ' B, if there are *-homomorphisms ' : A ! B and : B ! A* * for which O ' and ' O may be deformed by a path of endomorphisms to the identit* *y maps idA : A ! A and idB : B ! B, respectively. Two C*-algebras A and B are called `* *stably' homotopy equivalent if AK ' B K (this should not be confused with the notion of* * stable homotopy of spaces used in topology), so another way of stating the problem we * *consider is: when are C0(X) and C0(Y ) `stably' homotopy equivalent? To begin with we should remark that the analogous question for homotopy equi* *valence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X * *and Y are based homotopy equivalent. The situation for `stable' homotopy equivalence is more complicated and is c* *losely related to K-theory: for example, if two C*-algebras are `stably' homotopy equivalent t* *hen they have the same K-theoretic invariants. The main purpose of this paper is to show that* * the converse is not true: we give an example of two spaces X and Y which cannot be distingui* *shed by ________________________________ *the authors are partially supported by NSF grants 1 K-theory or even by connective K-theory but for which C0(X) K and C0(Y ) K ar* *e not homotopic. In order to do this we build up some machinery for analyzing such qu* *estions. The main result is Theorem 4.1 whose content is discussed below. Using a res* *ult of G. Segal, in its complex version (see [25]), we show that the homotopy type of C0(* *X) K can be approximated by the module structure of (complex) connective K-homology * *bu*X (or k*(X)) over its ring coefficient Z[u] ~=bu*. In the process we use the form* *alism of kk- groups of [11], to identify kk*(Y; X) with [*X; bu^ Y ] and we rely on a spectr* *al sequence discovered by Robinson [22] and improved by Elmendorf, Kriz, Mandell and May [1* *4]. To be specific, we show that if X and Y are finite connected CW complexes and the hom* *ological projective dimension of the bu*module bu*(X) is at most one, then C0(X)K ' C0(Y* * )K if and only if bu*(X) is isomorphic to bu*(Y ) as bu* modules. On the other ha* *nd we show that if the condition on the homological projective dimension is dropped, * *then there exists a pair of finite connected CW-complexes X and Y such that bu*X ~=bu *Y a* *s bu*- modules but C0(X) K 6' C0(Y ) K. (Recent work of Wolbert [30] shows that this phenomenon is related to the nonvanishing of a certain Bousfield -invariant. Th* *is point is discussed at the end of the paper.) >From what has been already said it is clea* *r that the `stable' homotopy equivalence for commutative C*-algebras fails to distinguish * *important topological invariants for the underlying spaces. In fact we show that there ex* *ists a pair of finite connected CW-complexes X and Y such that C0(X) K ' C0(Y ) K but X and Y are not stably homotopic, i.e., no iterated suspension iX is homotopic to iY* * . The group kk(Y; X) ~=[C0(X) K; C0(Y ) K] may be regarded as a connective version * *of the Kasparov group KK(C0(X); C0(Y )) which can be described in terms of homotopy cl* *asses of asymptotic morphisms as [[C0(X) K; C0(Y ) K]] (see [4], [10], [18]). Although* *, in general, there is no analog of the universal coefficient sequence of [26], (see however * *Corollary 4.3), one could use the above mentioned spectral sequence to compute kk*(Y; X) in ter* *ms of k*(X) and k*(Y ). A number of related results were announced in [7]. That announcement is now * *superseded by the present paper. For other applications of connective K-theory to C*-algeb* *ras we refer the reader to [25], [11], [9],[13], [6], [12] and [8]. The first named author thanks E. Effros for valuable discussions and A. Robi* *nson for some very helpful electronic correspondence. 2 The groups kk(X; Y ) For a compact space X with a base-point x0, let C0(X) denote the C*-algebra of * *continuous complex-valued functions on X which vanish at x0. If X and Y are spaces with ba* *se-point, 2 let Map(X; Y ) denote the set of continuous functions from X to Y which preserv* *e the base- point. The homotopy classes of these functions are denoted by [X; Y ]. If A and B are C*-algebras let Hom(A; B) denote the space of *-homomorphisms* * from A to B with the topology of pointwise-norm convergence. Its base point is the n* *ull homo- morphism. The homotopy classes [A; B] correspond to the path components of Hom(* *A; B). The tensor product C0(X) B is isomorphic to the C*-algebra of continuous funct* *ions from X to B which vanish at x0. One has C0(X) C0(Y ) ~=C0(X ^ Y ).The suspension o* *f a C*-algebra A is defined by A = C0(S1) A. If ' 2 Hom(A; B) then its suspension* * is defined by ' = idC0(S1) ' 2 Hom(A; B). This induces a map [A; B] ! [A; B]. For compact spaces X; Y with base points we define the semigroup kk(X; Y ) = [C0(Y ); C0(X) K] where K stands for the compact operators acting on a infinite dimensional separ* *able (com- plex) Hilbert space. The addition is defined by taking direct sum of *-homomorp* *hisms and identifying M2(K) with K. Using the suspension map we define the groups skkq(X; Y ) = lim kk(q+nX; nY ); q 2 Z: n!1 Proposition 2.1 Let i : A ,! X be a pair of finite connected CW complexes and * *let p : X ! X=A denote the quotient map. Then for any finite CW complex Y there are lon* *g exact sequences: p*s s : :!:skkn(Y; A) -i*!skkn(Y; X) -! kkn(Y; X=A) ! kkn-1(Y; A) ! : : : p*s i*s s : :!:skkn(X=A; Y ) -! kkn(X; Y ) -! kkn(A; Y ) ! kkn+1(X=A; Y ) ! : :* * : Since we are dealing with finite CW complexes, the mapping cone of C0(X) ! C0(A* *) is homotopic to C0(X=A). The two sequences correspond to the Puppe sequences for * *stable homotopy classes of *-homomorphisms of [25]. There is a natural identification (1) O : Map(X; Hom(A; B)) ! Hom(A; C0(X) B) O(f)(a)(x) = f(x)(a), f : X ! Hom(A; B), x 2 X and a 2 A. This induces a biject* *ion [X; Hom(A; B)] ! [A; C0(X) B]: 3 Let F (X) denote the space Hom(C0(X); K). In particular kk(Y; X) = [C0(X); C0(Y ) K] ~=[Y; Hom(C0(X); K)] ~=[Y; F (X)]: Throughout the paper we identify f with O(f). Let Mn denote the C*-algebra of * *n x n complex matrices and let F0(X) = [1n=1Hom(C0(X); Mn) with union taken by embedd* *ing Mn in Mn+1 by a 7! a 0. The natural inclusion F0(X) ! F (X) is a homotopy equi* *valence by [27, Proposition 1.2], hence kk(Y; X) ~=[Y; F0(X)]. G. Segal [27, Propositio* *n 1.5] showed that if A X is a pair of finite connected CW-complexes, then the natural map F* *0(X) ! F0(X=A) is a quasifibration with all the fibers homeomorphic to F0(A). This is * *a key result which leads to the following (see [11]). Theorem 2.2 Let X; Y be finite connected CW complexes. Then the suspension * *map in- duces an isomorphism kk(X; Y ) ! kk(X; Y ). A complete proof in given in [11, Corollary 3.1.8]. Here we just sketch the * *argument. Let CX denote the cone over X and Z the loop space of a based space Z. One checks t* *hat the inverse of the boundary map ffi in the homotopy exact sequence ! [Y; F0(CX)] ! [Y; F0(X)]!-ffi[Y; F0(X)] ! [Y; F0(CX)] ! associated with the quasifibration F0(CX) ! F0(X) with fiber F0(X) is given by * *the suspension map. Since F0(CX) is contractible the conclusion follows by exactnes* *s. QED The above theorem shows that if X; Y are finite connected CW complexes, then* * kk(X; Y ) is a group isomorphic to skk(X; Y ). Letting ( kk(qX; Y ); for q 0 kkq(X; Y ) = -q kk(X; Y ); for q 0; we have that kkq(X; Y ) is isomorphic to skkq(X; Y ). An important consequence* * of this isomorphism is that the exact sequences of Proposition 2.1 become available fo* *r the groups kkq(X; Y ). The groups kkq(X; Y ) were introduced in [11] as a connective vers* *ion of the Kasparov group KK(C0(Y ); C0(X)). They represent a very convenient framework fo* *r dealing with the multiplicative structure of connective K-theory. The relation to conne* *ctive K-theory will be explained in the next section. The groups kk* have a rich multiplicative structure. We begin by describing * *the multi- plication on kk. Given '0 2 Hom(C0(X); C0(Y ) K) and 0 2 Hom(C0(Y ); C0(Z) K), we define the 4 "composition" 0 '0 2 Hom(C0(X); C0(Z) K) to be the *-homomorphism idC0(Z) ( 0 idK)'0. '0 0idK idC0(Z) C0(X) -! C0(Y ) K ----! C0(Z) K K ------! C0(Z) K: The map is a fixed *-isomorphism whose specific choice is irrelevant as the au* *tomorphism group of K is path-connected. This composition induces a bilinear product kk(Y; X) kk(Z; Y ) ! kk(Z; X) ['0] [ 0] 7! [ 0 '0]: The bilinearity is a general fact [25], [16]. One checks that n 0 n'0 = n( 0 * * '0). Thus the diagram kk(Y; X) kk(Z; Y ) ! kk(Z; X) (2) # nn # n kk(nY; nX) kk(nZ; nY ) ! kk(nZ; nX) is commutative. Using the operation "00and the suspension functor we define a p* *roduct kk(pY; m X) kk(qZ; nY ) ! kk(p+qZ; m+n X); ['][ ] 7! ['][ ] = [p n']: The various maps involved in this definition are il* *lustrated in the diagram kk(pY; m X) kk(qZ; nY ) # np kk(p+nY; m+n X) kk(p+qZ; p+nY ) -! kk(p+qZ; m+n X); Since kkr(X; Y ) = limm kk(r+mX; m Y ), by using the commutativity of the diagr* *am 2, one checks that we obtain a well defined product kkr(Y; X) kks(Z; Y ) ! kkr+s(Z; X): In particular, T = kk*(S1; S1) is a ring which is seen to be isomorphic to Z[u]* * [11]. Here u has degree 2 and corresponds to the Bott element fi 2 Hom(C0(S1); C0(S3) K) * *which is a generator of ss3(U(1)) ~=[S3; F (S1)] ~=[C0(S1); C0(S3) K] ~=Z. Moreover * *k*(X) := q2Zkq(X), where kq(X) = limrkk(Sq+r; qX), is a T-module and we have a map : kk(Y; X) ! HomT(k*(Y ); k*(X)): 5 In section 3 we need the following description of the product structure of * *kk*. If we identify ' and with *-homomorphisms ' 2 Hom(C0(Sm ^ X); C0(Sp ^ Y ) K) and 2 Hom(C0(Sn ^ Y ); C0(Sq ^ Z) K), (u* *sing homeomorphisms of the type m X ~=Sm ^ X, etc.) then p n' will correspond to t* *he "composition" ' 2 Hom(C0(Sm ^ Sn ^ X); C0(Sp ^ Sq ^ Z) K) given by fl n m 1C0(Sn)' n p flidK (3) C0(Sm ^ Sn ^ X) -! C0(S ^ S ^ X) ------! C0(S ^ S ^ Y ) K ----! 1C0(Sp) idK p q id p q C0(Sp ^ Sn ^ Y ) K ---------! C0(S ^ S ^ Z) K K ---! C0(S ^ S ^ Z) K The maps fl are induced by flip homeomorphisms Sa ^ Sb ~=Sb ^ Sa. Let e 2 K be a one dimensional orthogonal projection. If A is a C*-algebra, * *let A : A ! A K be the map A(a) = a e. The unit of the ring kk*(X; X), denoted by 1, is g* *iven by the class of C0(X). Proposition 2.3 Suppose that X, Y are finite, connected CW complexes. Then C0(* *X) K is homotopy equivalent to C0(Y ) K if and only if there are ff 2 kk(X; Y ) and* * fi 2 kk(Y; X) such that fffi = 1 and fiff = 1. Proof. For C*-algebras A, B let : Hom(A; B K) ! Hom(A K; B K) denote the map ('0) = (idB ) O ('0 idK). It was shown in [29] that induces a bijec* *tion [A; B K] ! [A K; B K] and that ( 0 '0) is homotopic to ( 0) O ('0) for any '0 2 Hom(A; B K) and 0 2 Hom(B; C K). Moreover (A) is homotopic to idAK. The statement is obtained by applying these facts for A = C0(X) and B = C0(Y ).* * QED Theorem 2.4 Suppose that X, Y are finite, connected CW complexes. Then C0(X* *) K is homotopy equivalent to C0(Y ) K if and only if there is ff 2 kk(X; Y ) such* * that (ff) : k*(X) ! k*(Y ) is an isomorphism of groups. Proof. Suppose that C0(X) K is homotopy equivalent to C0(Y ) K and let ff and* * fi be given by Proposition 2.3. Then (ff) is an isomorphism with inverse (fi). Co* *nversely, suppose that (ff) is an isomorphism. From the very definition of kk(X; Y ) the* *re is ' 2 Hom(C0(Y ); C0(X) K) with ['] = ff. Then ' induces a map f : Hom(C0(X); K) ! Hom(C0(Y ); K), f( ) = '. At the level of homotopy groups this gives a map f* : ss*Hom(C0(X); K) ! ss*Hom(C0(Y ); K): 6 Since we can make the identification (ff) = f*, f* is an isomorphism, hence f i* *s a weak homotopy equivalence. Consequently, for any finite, connected CW complex Z, the* * map (fZ)* : [Z; Hom(C0(X); K)] ! [Z; Hom(C0(Y ); K)] induced by f is an isomorphism. Since (fZ)* can be identified with the map kk(Z; X) -xff-!kk(Z; X); by abstract nonsense, we obtain that there is fi 2 kk(Y; X) such that fffi = 1 * *and fiff = 1. We conclude the proof by applying Proposition 2.3. QED 3 The isomorphism [*X; bu ^ Y ] ! kk*(X; Y ) The reader is refered to [2] for the basic theory of spectra. Let BU denote t* *he spectrum of (reduced) complex K-theory. The spectrum of (reduced) complex connective K-* *theory will be denoted by bu. There is a map spectra bu ! BU such that ssr(bu) ! ss(B* *U ) is an isomorphism for r 0 and ssr(bu) = 0 for r < 0. These conditions determine bu u* *niquely up to a weak equivalence. Let X; Y be finite connected CW complexes. The Kasparov * *groups KK*(C0(Y ); C0(X)) are isomorphic to [*X; BU ^ Y ] [17]. It is then natural to* * consider the connective version of KK*(C0(Y ); C0(X)) which is [*X; bu^ Y ]. Remarkably,* * one can give these groups a very nice realization in terms of C*-algebras and homotopy * *classes of *-homomorphisms. To be specific, one shows that [X; bu^ Y ] ~=kk(X; Y ) ~=[C0(X); C0(Y ) K]: This isomorphism follows from a remarkable result of Segal [27] who identified * *bun with F (Sn) = Hom(C0(Sn); K). The structure map S1 ^ F (Sn) ! F (Sn+1) takes t ^ ' to ffit ' : C0(Sn+1) ~=C0(S1) C0(Sn) ! C K ~=K where ffit: C0(S1) ! C is the evaluation map ffit(a) = a(t). This result gives * *a nice realization of bu the (-)spectrum of complex connective K-theory in terms of homotopy class* *es of homomorphisms of C*-algebras. Let bu*(X) denote the reduced connective K-homolo* *gy and bu*(X) denote the reduced connective K-theory. If X is a finite connected CW-c* *omplex then we have isomorphisms k*(X) ~=bu*(X) and k*(X) ~=bu*(X). Recall that we den* *oted by fi 2 Hom(C0(S1); C0(S3) K) a morphism representing a generator of ss3(U(1))* * ~= 7 [S3; F (S1)] ~=[C0(S1); C0(S3) K] ~=Z. The Bott operation bun+2 ! bun is given* * by the map Hom(C0(Sn+2); K) ! Hom(C0(Sn); K) which sends ' to the composite Sn-1fi n+2 'idK C0(Sn) ----! C0(S ) K ----! K K -! K: As before the map is a fixed *-isomorphism whose specific choice is irrelevant* * as the automorphism group of K is arcwise connected. Let u : kn(X) ! kn+2(X) denote th* *e Bott operation. The ring structure of bu is induced by the multiplication : F (Sm )* * x F (Sn) ! F (Sm+n ), ' ^ 7! O (' ) as this is easily seen to be compatible with the * *multiplicative structure of BU . Recall that for a fixed Y , kkq(-; Y ) is a generalized cohomology theory. T* *he spectrum of this theory denoted by FY is given by the sequence of spaces F (nY ) = Hom(C0(n* *Y ); K). The structure map S ^ F (nY ) ! F (n+1Y ) is given by t ^ ' 7! ffit '. The dua* *l of this map F (nY ) ! F (n+1Y ) takes a *-homomorphism ' to its suspension '. Reca* *ll that the suspension map kk(X; Y ) ! kk(X; Y ) is an isomorphism. Thus FY is an spectrum. We want to show that bu ^ Y is equivalent to FY . To this purpose * *we define maps tn : F (Sn) ^ Y ! F (Sn ^ Y ), given by tn(' ^ y) = ' ffiy, where ffiy : * *C(Y ) ! C is the evaluation map at y. Proposition 3.1 The maps (tn) induce a weak equivalence of spectra o : bu ^ Y * * ! FY . Thus there is an isomorphism o* : [*X; bu^ Y ] ! kk*(X; Y ). Proof. The diagram S1 ^ F (Sn) ^ Y-1S^tn--!S1 ^ F (Sn ^ Y ) # # tn+1 n+1 F (Sn+1) ^ Y ---! F (S ^ Y ) is commutative. The vertical arrows are given by the structure maps. Therefor* *e o is a map of spectra. We need to show that o induces isomorphisms on the homotopy gr* *oups: ssq(bu ^ Y ) ! ssq(FY ) The above map can be identified with a map kq(Y ) ! kkq* *-1(S1; Y ). Since these are reduced homology theories it is enough to check the isomorphism* * for Y = S1. But this is certainly clear since for Y = S1, the map tn : F (Sn) ^ S1 ! F (Sn+* *1) coincides up to a flip with the structure map of bu. QED The ring structure of bu gives a multiplication [sY; bu^ X] [rZ; bu^ Y ] ! [s+rZ; bu^ X]: 8 On the other hand we have seen in the second section that the composition of *-* *homomorphisms gives rise to a multiplication kks(Y; X) kkr(Z; Y ) ! kks+r(Z; X) We are going to show that the isomorphism o* from Proposition 3.1 preserves the* * multiplica- tive structure. If we identify bun with F (Sn), then the product bum xbu n! bum+n is given b* *y (ff; fi) 7! O (ff fi) where ff fi 2 Hom(C0(Sm+n ); K K) and : K K ~=K. The product [sY; bu^ X] [rZ; bu^ Y ] ! [s+rZ; bu^ X]: ! j 7! !j is defined as follows. If ! and j are represented by f 2 Map(Sp ^ Y; F (Sm ) ^ X); andg 2 Map(Sq ^ Z; F (Sn) ^ Y ) with r = p - m and s = q - n, then !j is represented by the composite g * f: 1Sp^g p n fl n p 1F(Sn)^f (4) Sp ^ Sq ^ Z ----! S ^ F (S ) ^ Y -! F (S ) ^ S ^ Y -----! ^1X n m fl m n F (Sn) ^ F (Sm ) ^ X ---! F (S ^ S ) ^ X -! F (S ^ S ) ^ X The maps fl correspond to various flip homeomorphisms. Proposition 3.2 With f and g as above, tm+n O (g * f) = (tn O g) (tm O f). Proof. Recall that tm : F (Sm ) ^ X ! F (Sm ^ X) maps an element ff ^ x to ff * * ffix 2 Hom(C0(Sm ^ X); K) and tm O f : Sp ^ Y ! F (Sm ^ X) is identified via O with a* * *- homomorphism ' 2 Hom(C0(Sm ^ X); C0(Sp ^ Y ) K). Similarly we identify tn O g * *with some 2 Hom(C0(Sn ^ Y ); C0(Sq^ Z) K) and tm+n O (g * f) with some ss 2 Hom(C* *0(Sm ^ Sn ^ X); C0(Sp^ Sq^ Z) K). Let a b c 2 C0(Sm ) C0(Sn) C0(X) ~=C0(Sm ^ Sn ^* * X) be an elementary tensor. It suffices to show that (5) ss(a b c) = ( ')(a b c) when both sides are evaluated at any point t ^ z where t 2 Sp and z 2 ^Sq ^ Z. * *In order to fix the notation let's say that (6)g(z) = fi ^ y 2 F (Sn) ^ Y; (1F(Sn)^ f)(fi ^ t ^ y) = fi ^ ff ^ x 2 F (Sn) * *^ F (Sm ) ^ X 9 Then using (4) we compute (g * f)(t ^ z): (7) t ^ z 7! t ^ fi ^ y 7! fi ^ t ^ y 7! fi ^ ff ^ x 7! ( O (fi ff) ^ x) O fl * *= (g * f)(t ^ z) Therefore (8) ss(a b c)(t ^ z) = ( O (fi ff) ffix)(b a c) = (fi(b) ff(a))c(x) In order to compute the right-hand side of (5) we show first that, with the not* *ation as in (6) (9) (1C0(Sp) idK)fl(b w)(t ^ z) = fi(b) w(t ^ y) 2 K K for all b 2 C0(Sn) and w 2 C0(Sp^ Y ) K. The map fl is as in (1.3). Since the * *both sides of (8) depend linearly and continuously on w, we may assume that w is an elementar* *y tensor w = u v 2 C0(Sp) C0(Y ) K. We have fl(b u v ) = u b v , and using (6) (1C0(Sp) idK)(u b v )(t ^ z) = u(t) (b v)(z) = u(t) fi(b) v(y) = fi(b) u(t)v(y) = fi(b) w(t ^ y) We have identified C K ~= K C ~=K. After this preparation we are able to com* *pute ( ')(a b c)(t ^ z) using (2) and (6). We have (1C0(Sp) idK)t^z a b c 7! b a c 7! b '(a c) ------------! fi(b) '(a c)(t ^ y) (by (8) with w = '(a c)) = fi(b) ff(a) c(x) 7! (fi(b) ff(a))c(x) Combining (8) and (10) we see that (5) holds true and this concludes the proof * *of the Proposition. QED As a consequence of Proposition 3.2 we obtain the following. Corollary 3.3 The map o* : [*X; bu^ Y ] ! kk*(X; Y ) preserves the products. Corollary 3.4 Let X; Y be finite connected CW complexes. Then C0(X) K is homot* *opy equivalent to C0(Y ) K if and only if the image of the natural map : [X; bu^ Y ] ! Hombu*(bu*X; bu*Y ) contains an isomorphism. 10 Proof. By Proposition 3.1 and Corollary 3.3, there is a commutative diagram kk(X,Y) -o*! [X; bu^ Y ] # # HomT(k*(X); k*(Y )) -o*! Hombu*(bu*X; bu*Y ) with both horizontal arrows being bijections. We conclude the proof by applying* * Theorem 2.4 . QED 4 The main result Let bu* denote ss*bu ~=Z[u], where u is the Bott operation and has degree 2. Si* *nce Z[u] has global dimension two, the homological projective dimension of the bu* module bu* **(X) is at most two. In this section we will begin the proof of the following: Theorem 4.1 (a) Let X and Y be finite connected CW complexes. Suppose that* * the homological projective dimension of the bu* module bu*(X) is at most one. * * Then C0(X) K ' C0(Y ) K if and only if bu*(X) is isomorphic to bu*(Y ) as bu** * mod- ules. (b) There exists a pair of finite connected CW-complexes X and Y such that bu*X* * ~=bu*Y as bu*-modules but C0(X) K 6' C0(Y ) K. (c) There exists a pair of finite connected CW-complexes X and Y (each having t* *wo cells) such that C0(X) K ' C0(Y ) K but X and Y are not stably homotopy equival* *ent. It has been shown in previous sections that C0(X) K ' C0(Y ) K if and only* * if the image of the natural map : [X; bu^ Y ] ! Hom bu*(bu*X; bu*Y ) contains an isomorphism. Thus to prove part (a) we will show that is surjecti* *ve, to prove part (b) we will show that while Hom bu *(bu*X; bu*Y ) contains isomorphi* *sms, no isomorphism is in the image of , and to prove part (c) we will show that there * *is an isomorphism in the image of . Our main tool for doing this will be the following spectral sequence, which * *was discovered by Robinson [22] and improved by Elmendorf, Kriz, Mandell and May [14, Theorem * *IV.3.1]. 11 Theorem 4.2 (a) There is a spectral sequence Er(X; Y ) which converges to [-s* *-tX; bu^ Y ] and is natural in X and Y . (b) E2(X; Y ) = Exts;tbu*(bu*X; bu*Y ); and the edge homomorphism [X; bu^ Y ] ! E0;01,! E0;02= Hom bu*(bu*X; bu*Y ) is equal to . (c) Es;t2= 0 for s > 2. (d) The only possible nonzero differential is d2 : Hom tbu*(bu*X; bu*Y ) ! Ext2;t-1bu*(bu*X; bu*Y ) Here Exts;tmeans the part of Extswhich has internal degree t; for example Ex* *t0;t= Hom t denotes the homomorphisms of graded groups which lower degree by t. The filtra* *tion of [*X; bu^ Y ] is described in the last section of the paper. We have exact seque* *nces: (10) 0 ! F s+1[-s-tX; bu^ Y ] ! F s[-s-tX; bu^ Y ] ! Es;t1! 0 and F s[-s-tX; bu^ Y ] = 0 for s > 2. Corollary 4.3 Let X and Y be finite connected CW complexes. Suppose that the ho* *mological projective dimension of the bu* module bu*(X) is at most one. Then there is an* * exact sequence (11) 0 ! Ext1;-1bu*(bu*X; bu*Y ) ! [X; bu^ Y ] -! Hom bu*(bu*X; bu*Y ) ! 0 Proof. The assumption on the homological projective dimension of bu*X implies t* *hat Exts;t(bu*X; bu*Y ) = 0 for s > 1. Thus all the d2 differentials are zero and * *the edge homomorphism is surjective. The statement follows now from (10). QED Remark Notice that if we let X and Y be the spaces described in part (b) of * *4.1 then d2 6= 0 in the spectral sequence Er(X; Y ). It will become apparent from our ar* *guments below 12 that if X, Y are finite connected CW complexes, then C0(X) K ' C0(Y ) K if an* *d only if there is an isomorphism ff 2 E0;02= Hom bu*(bu*X; bu*Y ) with d2(ff) = 0. The statement of Theorem 4.2 does not agree precisely with the corresponding* * statements in [22] and [14]; we shall explain how to deduce it from the work of those auth* *ors in section 6, except that we will show part (c) in section 5. It is now easy to complete the proofs of part (a) and (c) of Theorem 4.1. Pa* *rt (a) is a straightforward consequence of Corollary 4.3 and Corollary 3.4. Next we deal with part (c). Let ff : Sm ! Sn be any map which is not stably * *trivial. Let X be the cofiber of ff and let Y = Sm+1 _ Sn (i.e., Y is the cofiber of the tri* *vial map from Sm to Sn). The stable maps from Sm to Sn form a finite group, but bu*Sn is eith* *er zero or torsion free, so ff must induce the zero map of bu-homology. Now the long exact* * sequence for bu*X shows at once that bu*X ~=bu*Sm+1 bu*Sn ~=bu*Y: In particular, bu*X is a free bu*-module, and so we have that Exts;t(bu*X; bu*Y* * ) = 0 for s > 0. Thus the spectral sequence collapses and is an isomorphism, implying C0* *(X)K ' C0(Y ) K. On the other hand, if X were stably homotopic to Y then the compos* *ite of stable maps Sm -ff!Sn ,! X -'!Y ! Sn (where the last map is the projection of Y on its wedge-summand Sn) would be nu* *llhomo- topic (since the first two of these maps are consecutive maps in a cofiber sequ* *ence), while the composite of the last three would be a stable equivalence of Sn (since it i* *s clearly an isomorphism in homology). This would imply that ff is stably nullhomotopic, con* *trary to our initial assumption. We remark that if ff is chosen to be the nontrivial map j : Sn+1 ! Sn then a* *n easy calculation shows that KO*X is not isomorphic to KO*Y . Thus in this case we h* *ave an example where C0(X) K is homotopy equivalent to C0(Y ) K over the complex num* *bers but not over the reals. We now turn to part (b) of Theorem 4.1. Fix an odd prime p and an integer k * * 2, and let M be defined by the cofiber sequence p k Sk!- S ! M: By [1, Theorem 1.7] and [5], there is a map A : 2p-2M ! M 13 which induces an isomorphism in K-homology. Let N be defined by the cofiber seq* *uence 2p-2M -A!M ! N: Let f : N ! 2p-1M be the next map in this cofiber sequence, and let g : 2p-1M !* * S2p+k be the next map in the cofiber sequence defining M. Finally, let h = g O f, le* *t X be the cofiber of h, and let Y = N _ S2p+k (i.e., Y is the cofiber of the trivial map * *from N to S2p+k). (It is not difficult to show that X is homotopic to a 3-cell complex, but we* * shall not need to know this). In order to verify the first assertion of part (b) we begin by noting that b* *uk+nM is Z=p if n is a nonnegative even integer and 0 otherwise. The long exact sequence for* * bu*N now shows that buk+n(N) is Z=p if n is a nonnegative even integer less than 2p - 2,* * and zero otherwise. Hence h induces the zero map of bu-homology, and we conclude that bu *X ~=bu*N bu*S2p+k~= bu*Y as bu*-modules. Before proceeding, we need to introduce some notation. Given any spectra W a* *nd Z and a map ff : W ! bu^ Z, let "ffdenote the composite bu ^ W -1^ff-!bu^ bu^ Z -! bu ^ Z; where is the multiplication of the ring spectrum bu. Note that (ff) is the map* * of homotopy groups induced by "ff. Also, given a map fi : W ! Z, let fi0 denote the composi* *te fi 0 j^1 W -! Z = S ^ Z --! bu^ Z; where j is the unit map of the ring spectrum bu. The proof of the remaining assertion of part (b) will be by contradiction, s* *o suppose that C0(X) K ' C0(Y ) K. Then there is a map ff : X ! bu ^ Y for which (ff) i* *s an isomorphism; this implies that "ffinduces an isomorphism of homotopy groups. No* *w consider the following composite, which we shall denote by : 1^j 2p+k bu ^ S2p+k-1^i-!bu^ X -"ff!bu^ Y --! bu^ S (here i is the inclusion of S2p+k in X, and j is the projection of Y to its wed* *ge-summand S2p+k). The map induces an isomorphism of homotopy groups, since (1^i)* and (1* *^j)* are isomorphisms in all dimensions where ss*(bu ^ S2p+k) is nonzero. But O h0is ho* *motopically 14 trivial, since h and i are consecutive maps in a cofiber sequence. It follows t* *hat h0is trivial. To establish a contradiction, we shall show that h0is nontrivial. For this purpose we need to use a product pairing in the spectral sequence o* *f Theorem 4.2. Given ff : V ! bu^W and fi : W ! bu^Z let us define the "composition" fi *ff 2 * *[V; bu^Z] to be the composite 1^fi ^1 V -ff!bu^ W --! bu ^ bu^ Z --! bu^ Z: (Note that this composition operation corresponds to ordinary composition of ho* *momor- phisms of C* algebras). The following result is due to Robinson [24] and Elmen* *dorf, Kriz, Mandell and May [14, Theorem IV.3.4]; we state it in a somewhat different form * *which we shall reconcile with their version in section 6. Theorem 4.4 There is a pairing of spectral sequences Er(V; W ) Er(W; Z) ! Er(V; Z) which is the Yoneda product ([19, Section 3.5]) when r = 2 and is induced by th* *e * operation when r = 1. Notice that h0= g0* f0. To complete the proof that h0is nontrivial, we shall* * need the following facts. Proposition 4.5 (a) E1;-12(N; 2p-1M), E1;-12(2p-1M; S2p+k) and E2;-22(N; S2p+k* *) are each isomorphic to Z=p. (b) f0 and g0 are represented by nontrivial elements x 2 E1;-12(N; 2p-1M), an* *d y 2 E1;-12(2p-1M; S2p+k). (c) The Yoneda product of x and y is a nontrivial element of E2;-22(N; S2p+k). We shall prove parts (a) and (c) in section 5, and part (b) in section 6. As* *suming these for the moment, we see that h0= g0* f0 is represented by a nontrivial element o* *f E2;-22. If h0 were trivial then this element of E2 would have to be hit by a differential.* * But such a differential would have to originate in Hom bu*(bu*N; bu*S2p+k), and this Hom * *group is zero because there are no nontrivial homomorphisms from Z=p to Z. 15 5 Algebraic calculations in the E2-term. In this section we prove parts (a) and (c) of Proposition 4.5. The following re* *sult will allow us to calculate the relevant Ext groups. Theorem 5.1 Let A and B be any two bu*-modules. Let u denote the generator of* * ss2bu, and let u1 : A ! A and u2 : B ! B denote multiplication by u. Then there is a l* *ong exact sequence u*1-u2* t-2 0 ! Hom tbu*(A; B)! Hom tZ(A; B) ----! Hom Z (A; B) ! u*1-u2* 1;t-2 Ext1;tbu*(A; B)! Ext1;tZ(A; B) ----! ExtZ (A; B) ! Ext2;tbu*(A; B)! 0 and Exts(A; B) = 0 for s > 2. From now on we will abbreviate bu* by T . Proof of Theorem 5.1. Let u3 : T ! T be multiplication by u, and let : T A* * ! A be the action of T on A. It is easy to check that the sequence 0 ! T A -1u1-u31------!T A -! A ! 0 is an exact sequence of T -modules in which the first map raises degrees by 2. * *It therefore induces a long exact sequence of Ext-groups: (1u1)*-(u31)* t-2 @ (12) 0 ! Hom tT(A; B)! Hom tT(A; B) ----------! Hom T (T A; B) -! (1u1)*-(u31)* 1;t-2 @ Ext1;tT(A; B)! Ext1;tT(T A; B) ----------! ExtT (T A; B) -! Ext2;tT(A; B)! . . . But a standard change of rings theorem [3, Proposition VI.4.1.3] says that the * *composite Ext *T(T A; B) ! Ext*Z(T A; B) ! Ext*Z(A; B) is an isomorphism (here the first map is the evident forgetful map and the seco* *nd is induced by the Z-module map which takes m to 1 m). Using this isomorphism, and the fac* *t that ExtsZ= 0 for s > 2, it is easy to see that in the long exact sequence we have j* *ust given all terms are zero after Ext2;tT(A; B) and that the initial part of the sequence is* * isomorphic to that given in the theorem. QED 16 We can now prove part (a) of Proposition 4.5. First we consider Ext1;-1T(bu*N; bu*2p-1M): Let A denote the graded group bu*N and let B denote bu*2p-1M. We have seen in t* *he previous section that Ak+n is Z=p if n is even with 0 n < 2p - 2 and 0 otherwi* *se, while B2p-1+k+nis Z=p when n is a nonnegative even integer and 0 otherwise. This is d* *escribed in the folowing table, the last row of which will be used in the second part of* * the proof. n 0 1 2 3 4 . . .2p - 4 2p - 2 2p - 1 2p 2p + 1 2p + 2 . .* * . Ak+n Z=p 0 Z=p 0 Z=p . . .Z=p 0 0 0 0 0 0 . .* * . Bk+n 0 0 0 0 0 . . . 0 0 0 Z=p 0 Z=p 0 . .* * . Ck+n 0 0 0 0 0 . . . 0 0 0 0 Z 0 Z . .* * . By definition, Hom tZ(A; B) is 1Y Hom Z(Ai; Bi-t); i=-1 so in our case Hom -1Z(A; B) is zero, while Hom -3Z(A; B) is Z=p, so the cokern* *el of u*1- u2*is Z=p. Since Ext1;-1Z(A; B) is zero we conclude that Ext1;-1T(A; B) is Z=p as req* *uired. Next we consider Ext1;-1T(bu*2p-1M; bu*S2p+k). Let B be as above and let C * *denote bu*S2p+k. Then C2p+k+nis Z if n is an even integer 0 and 0 otherwise. It fol* *lows that Hom -3Z(B; C) = 0, so Ext1;-1T(B; C) is the kernel of u*1-u2* 1;-3 Ext1;-1Z(B; C) ----! ExtZ (B; C): Now an element of either Ext1;-1Z(B; C) or Ext1;-3Z(B; C) is a sequence x2p+k-1; x2p+k+1; x2p+k+3; : : : of elements of Z=p, and (u*1- u2*)(x2p+k-1; x2p+k+1; x2p+k+3; : :):= (x2p+k+1- x2p+k-1; x2p+k+3- x2p* *+k+1; : :):: Therefore the kernel of u*1- u2*is a copy of Z=p generated by (1; 1; 1; : :):. Finally, consider Ext2;-2T(A; C). We have Ext1;-2Z(A; C) = 0 and Ext1;-4Z(A;* * C) = Z=p, so we conclude that Ext2;-2T(A; C) = Z=p. QED 17 We conclude this section by proving part (c) of Proposition 4.5. We continue* * with the notation of the previous proof. From what has been shown we know that the bound* *ary map Hom -3T(T A; B) -@!Ext1;-1T(A; B) is an isomorphism, so there is an element x0of Hom -3T(T A; B) with @(x0) = x.* * By [19, Theorem III.9.1] we see that the boundary operator @ is (up to sign) the Yoneda* * product with a certain element, and in particular we have xy = (@x0)y = @(x0y). Since Ext1;-4T(T A; C) -@!Ext2;-2T(A; C) is also an isomorphism, it suffices to show that x0y 6= 0. Next consider the diagram Hom -3T(T A; B) Ext1;-1T(B; C) ! Ext1;-4T(T A; C) # Iae # I Hom -3Z(A; B) Ext1;-1Z(B; C) ! Ext1;-4Z(A; C) # r1 r2 # r3 Hom Z(Ak+2p-4; Bk+2p-1) ExtZ(Bk+2p-1; Ck+2p)! ExtZ(Ak+2p-4; Ck+2p) Here the horizontal arrows are Yoneda products (which in this case are simply c* *omposition operations), ae is the evident restriction map, and I denotes the isomorphism u* *sed in the proof of Theorem 5.1; from the description of I given there, it is easy to see * *that the upper square commutes. The lower square clearly commutes, and the bottom horizontal * *map is an isomorphism. Also, the restriction maps r1 and r3, and the composite r2 O ae* *, have been shown to be isomorphisms in the proof just given. The result follows. QED 6 Detection of elements in filtration 1. In this section we shall prove part (b) of Theorem 4.5. Let W and Z be any spaces, and let fi : W ! Z be any map. As in section 1, w* *e write fi02 [W; bu^ Z] for the composite fi 0 j^1 W -! Z = S ^ Z --! bu^ Z; 18 and we recall that (fi0) is the homomorphism induced by fi in bu-homology. Now we assume that fi induces the zero homomorphism in bu* homology. (for ex* *ample, this is true for the maps f and g in Proposition 4.5). Then (fi0) = 0, so by p* *art (b) of Theorem 4.2, fi0 determines an element (which may be zero) in E1;-11(W; Z) and by parts (b) and (d) of Theorem 4.2 this group is equal to Ext1;-1T(bu*W; bu*Z); where T denotes ss*bu. Now by [19, Theorem III.6.4] an element of this Ext-grou* *p is identified with an equivalence class of extensions of T -modules 0 ! bu*Z ! A ! bu*W ! 0; (where the map A ! bu*W raises degrees by 1) so in particular the map fi is rep* *resented by a class of such extensions. Our aim is to describe explicitly an extension whic* *h represents fi. To do this, we let fi W -! Z ! C(fi) ! W be the cofiber sequence determined by fi. Applying bu*, and using the fact that* * fi* = 0, we get a short exact sequence of T -modules (13) 0 ! bu*Z ! bu*C(fi) ! bu*W ! 0 where the map bu*C(fi) ! bu*W raises degrees by 1. Theorem 6.1 The extension (13) represents the element of Ext1;-1T(bu*W; bu*Z)* * deter- mined by fi. In particular, this element of Ext is zero if and only if (13) is* * split as a short exact sequence of T -modules. Before proving this, we use it to complete the proof of Proposition 4.5(b). * * >From the definition of f, we see that the cofiber sequence determined by f has the form f 2p-1 N -! M ! M ! N The short exact sequence induced by this is 0 ! bu*2p-1M ! bu*M ! bu*N ! 0; 19 and this is certainly not split since there is no nontrivial T -module homomorp* *hism from bu*N to bu*M (note that the generator u of T acts nilpotently on the former but* * not on the latter). The argument for g is similar and is left to the reader. In the remainder of the section we give the proof of Theorem 6.1. It is nece* *ssary first to put this result in a broader context by relating it to the work of Elmendorf, K* *riz, Mandell and May. Let R be an A1 ring spectrum (for our purposes it is not necessary to know * *what this means, we only need to know that buis one; in fact it has the stronger structur* *e of an E1 ring spectrum, by [20, VIII.2.1]). Then Elmendorf, Kriz, Mandell and May, following * *earlier work of Robinson [21, 22, 23, 24], show how to define a category of R-module spectra* * in which one can do homotopy theory (technically, these are A1 R-modules, which is a strict* *er notion than the R-module spectra used in [2], for instance). The homotopy category is* * denoted DR. For example, when R is the sphere spectrum S, DR is the usual stable catego* *ry. An R-module is a spectrum with extra structure, so there is a forgetful functor wh* *ich takes an R-module to its underlying spectrum; there is also a "free R-module" functor F * *which is left adjoint to the forgetful functor. Thus if we write [A; B]R for the set of homot* *opy classes in DR when A and B are two R-modules, then we have [F(X); B]R = [X; B] whenever B is an R-module. In particular, we can let B = F(Y ); the underlying * *spectrum of B is then R ^ Y , by [14, Proposition III.1.3], and so we have [F(X); F(Y )]R = [X; R ^ Y ]: Now there is a spectral sequence Ext s;tss*R(ss*A; ss*B) =) [-s-tA; B]R; (see [22], [14, Theorem IV.3.1]), and letting R = bu, A = F(X) and B = F(Y ) gi* *ves the spectral sequence of Theorem 4.2. Properties (a) and (b) of Theorem 4.2 are im* *mediate from the corresponding properties in [22] and [14, Theorem IV.3.1]. Similarly, Theorem 6.1 is immediate from the following more general result: Theorem 6.2 Let fl A -! B ! C ! A ! be a cofiber sequence of R-modules, with ss*fl = 0. Then fl is in F 1[A; B]R, a* *nd the image of fl under the composite F 1[A; B]R=F 2[A; B]R = E1;-11(A; B) ,! Ext1;-1ss*R(ss*A; ss*B) 20 is the element represented by the short exact sequence 0 ! ss*B ! ss*C ! ss*A ! 0: If f : M ! N is a map of graded modules, then f : M ! N is defined by (f)i = fi-1where M is the module (M)i = Mi-1. In order to prove Theorem 6.2 we have to recall how the spectral sequence is defined; we use the construction in* * [14, Section IV.4] and the reader is referred to that source for further information. First * *pick a resolution of ss*A by free ss*R-modules: d-1 (14) . . .M2 d2-!M1 d1-!M0 d0-!M-1 = ss*A --! 0 For each i 0 let Xi be a wedge of spheres indexed by a basis for Mi. Define R* *-module spectra Aifor i 0, with A0 = A, inductively by the cofiber sequences ji+1i+1 iFXi-ki!Ai! Ai+1--! FXi; with the following properties: (i)kiinduces an epimorphism ss*iFXi! ss*Aifor i 0. (ii)ss*Ai~= i(ker di-1) for i 0. (iii)kirealizes idi: iMi! i(ker di-1) on ss* for i 0. (iv)ji+1realizes the inclusion i+1(ker di) ! i+1Mion ss* for i 0. Observe that (ii) together with (iii) implies (i). Then the groups Dp;q1= [-p-qAp; B]R and Ep;q1= [-qFXi; B]R form an exact couple, which in the usual way induces the required spectral sequ* *ence. To prove that the E2-term has the desired form, one uses the fact that [FXi; B]R ~=[Xi; B] ~=Hom ss*R(Mi; ss*B): It is shown in [14, Section IV.4] that the spectral sequence is independent, fr* *om E2 on, of the choices made in this construction. 21 Now let fl : A ! B be a map of R-modules which induces the zero homomorphism* * of homotopy groups. Since X0 is a wedge of spheres, the composite fl X0 ! A -! B is nullhomotopic, and since F is left adjoint to the forgetful functor the comp* *osite fl FX0 ! A -! B is also nullhomotopic; thus it is possible to extend fl to a map : A1 ! B: The composite FX1 k1-!A1 -! B induces a map of homotopy groups fl: M1 = ss*FX1 ! ss*B which raises degrees by 1 and is a ss*R-homomorphism. Using (ii)-(iv) we see th* *at j1*k1*d2 = d1d2 =.0Since j1*is injective, this implies that fld2 = *k1*d2 = 0: Since (14) is a projective resolution, and fld2 = 0, flrepresents an element of* * Ext1;-1R(ss*A; ss*B), and by the definition of the spectral sequence this is the element correspondin* *g to fl. It remains to show that flcorresponds to the extension 0 ! ss*B ! ss*C ! ss*A ! 0: Since the sequence ss*FX1 ! ss*FX0 ! ss*A ! 0 is a partial projective resolution, Theorem III.6.4 of [19] tells us that it su* *ffices to show that there is a commutative diagram fl ss*FX1 -! ss*B # # (15) ss*FX0 ! ss*C # # ss*A -=! ss*A: 22 To construct such a diagram, we first observe that the composite fl FX0 k0-!A -! B is nullhomotopic, and hence k0 lifts to -1C, where C is the cofiber of fl. Thus* * we have a homotopy-commutative diagram FX0 ! -1C # k0 # A -=! A Now it is a well-known fact of homotopy theory (see for instance [28, Lemma 8.3* *1]) that any homotopy-commutative diagram of the form X -ff!X0 # f # f0 fi 0 Y -! Y extends to a homotopy commutative diagram X -ff! X0 # f # f0 fi 0 Y -! Y # # Cf ! Cf0 # # X -ff-! X0 # f # f0 fi 0 Y --! Y where the columns are the cofiber sequences of f and f0. In our case, we obtain* * a homotopy- commutative diagram FX0 ! -1C # k0 # A -=! A # # (16) A1 ! B # # FX0 ! C # # A -=! A 23 The map A1 ! B in this diagram is a candidate for the map mentioned above, * *and so the composite (k1)* ss*FX1 ---! ss*A1 ! ss*B represents fl. Now applying ss* to the bottom half of diagram (17) and precompo* *sing with (k1)* gives the diagram (16), and this concludes the proof. Remark The filtration of [*A; B]R in the above spectral sequence is defined * *by letting F i[*A; B]R to be the image of the map [*Ai; B]R ! [*A; B]R induced by the evident iterate A ! Ai. If R = bu, since bu* = Z[u] has homolog* *ical projective dimension equal to 2, one can find a free resolution (14) such that * *Mi = 0 for i 3. It follows from (ii) that ss*Ai = 0 for i 4 so that the filtration vani* *shes in all the degrees greater than three. For the spectral sequence from Theorem 4.2 we * *see that F s[*X; bu^ Y ] = 0 for s > 2 since Es;t1= 0 for s > 2. Remark Recently, Wolbert [30] has shown that there is a Bousfield -invariant* * A 2 Ext2;-1bu*(ss*A; ss*A) with the differential d2 : Hom bu*(ss*A; ss*B) ! Ext2;-1* *bu*(ss*A; ss*B) given by d2f = Af - fB. Suppose that X, Y are finite connected CW complexes and let A = bu^ X and B = bu^ Y . Therefore with X := bu^X 2 Ext2;-1bu*(bu*X; bu*Y ) we* * have that C0(X) K is homotopy equivalent to C0(Y ) K if and only if there is an is* *omorphism f 2 Hom bu*(bu*X; bu*Y ) such that fX = Yf. 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