PRODUCT AND OTHER FINE STRUCTURE IN POLYNOMIAL RESOLUTIONS OF MAPPING SPACES STEPHEN T. AHEARN AND NICHOLAS J. KUHN Abstract.Let MapT(K, X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite compl* *ex, Greg Arone has recently given an explicit model for the Goodwillie tower* * of the functor sending a space X to the suspension spectrum 1 MapT(K, X). Applying a generalized homology theory h* to this tower yields a spect* *ral sequence, and this will converge strongly to h*(MapT (K, X)) under suita* *ble conditions, e.g. if h* is connective and X is at least dimK connected. E* *ven when the convergence is more problematic, it appears the spectral sequen* *ce can still shed considerable light on h*(MapT (K, X)). Similar comments h* *old when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to usefu* *l in- teresting additional structure in the associated spectral sequences. For* * exam- ple, the diagonal on MapT(K, X) induces a `diagonal' on the associated t* *ower. After applying any cohomology theory with products h*, the resulting spe* *c- tral sequence is then a spectral sequence of differential graded algebra* *s. The product on the E1 -term corresponds to the cup product in h*(MapT (K, X)) in the usual way, and the product on the E1-term is described in terms of group theoretic transfers. We use explicit equivariant S-duality maps to show that, when K is the sphere Sn, our constructions at the fiber level have descriptions in ter* *ms of the Boardman-Vogt little n-cubes spaces. We are then able to identify, i* *n a computationally useful way, the Goodwillie tower of the functor from spe* *ctra to spectra sending a spectrum X to 1 1 X. 1.Introduction Let MapT (K, X) denote the mapping space of continuous based maps between two based spaces K and X. To compute its homology or cohomology with re- spect to any generalized theory, it suffices to consider the suspension spectrum 1 MapT (K, X)+ , where Z+ denotes the union of a space Z with a disjoint base- point. If one fixes K and lets X vary, one gets a functor from spaces to spectra. As- suming, as we will also do from now on, that K is a finite CW complex, G. Arone [Ar] has recently studied this functor from the point of T. Goodwillie's calcul* *us of functors [G1 , G2, G3]. He defines a very explicit natural tower P K(X) of fibr* *ations ____________ Date: August 9, 2001. 2000 Mathematics Subject Classification. Primary 55P35; Secondary 55P42. Research was partially supported by the N.S.F.. 1 2 AHEARN AND KUHN of spectra under 1 MapT (K, X)+ , .. . | | fflffl| P2K(X)88 qqqqq | qqq | qqqq fflffl| qqqqqq hh3P1K(X)3h qqqq hhhhhhhhh | qqqq hhhhh | qq hhhh fflffl| 1 MapT (K, X)+ ________________//_P0K(X), and shows that the connectivity of the maps eKk(X) : 1 MapT (K, X)+ ! PkK(X) increases linearly with k as long as the dimension of K is no more than the con- nectivity of X. The kth fiber FkK(X) of the tower is shown to be naturally weak* *ly equivalent to a homotopy orbit spectrum: (1.1) FkK(X) ' MapS (K(k), X^k)h k. Here K(k)= K^k= k(K), the quotient of the k-fold smash product K^k by the fat diagonal k(K), and MapS (K(k), X^k) denotes the spectrum of stable maps from K(k)to X^k, a spectrum with an action of the kth symmetric group k. Since this is a homogeneous polynomial functor of degree k, Arone has identified the Goodwillie tower of 1 MapT (K, X)+ . Applying a generalized homology theory h* to this tower yields a (left half p* *lane) spectral sequence, and this will converge strongly to h*(MapT (K, X)) under sui* *t- able conditions, e.g. if h* is connective and X is at least dim K connected. Ev* *en when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h*(MapT (K, X)). Similar comments hold when a cohomology theory is applied. When K is the circle S1, and the homology theory is ordinary, one can show th* *at the resulting spectral sequence is the classical Eilenberg-Moore spectral seque* *nce. For other K, it appears that the Arone spectral sequences are organized more usefully than the older Anderson spectral sequence [An ] for computing the homo* *logy and cohomology of MapT (K, X).1 For the deepest applications of essentially any interesting spectral sequence* *, one uses additional structure that the spectral sequences carries. It is the purpos* *e of this paper to study various geometric properties of the towers P K(X) which lead to * *such interesting additional structure in their associated spectral sequences. For ex* *ample, we construct a `diagonal' on P K(X). After applying any cohomology theory with products h*, the resulting spectral sequence will be then be a spectral sequenc* *e of differential graded algebras. The product on the E1 -term will correspond to the cup product in h*(MapT (K, X)) in the usual way, and the product on the E1-term will be described in terms of group theoretic transfers. ____________ 1We note that [BG ] suggests that the two spectral sequences are related. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 3 Perhaps the towers of greatest interest are those when K = Sn, the n-sphere. We combine (1.1) with an explicit unstable k-equivariant S-duality map ffi(n, k) : C(n, k)+ ^ Sn(k)! Snk, to construct an explicit natural weak homotopy equivalence n nk ^k (1.2) FkS (X) ' (C(n, k)+ ^ MapS (S , X ))h k. Here C(n, k) is the Boardman-Vogt space of k disjoint little n-cubes in a big n* *-cube [M ]. In terms of the extended power constructions of [LMMS ], this last equivalen* *ce yields a weak equivalence n -n ^k (1.3) FkS (X) ' C(n, k)+ ^ k ( X) . Here -n X denotes the nthdesuspension of the suspension spectrum of X. Using either (1.2) or (1.3), our generalnstructure theorems for P K(X) simpli* *fy in nice ways when specialized to P S (X). This leads to the spectral sequences * *for computing h*( nX) having lots of extra algebraic structure that can be related * *to classical calculations, together with statements about how these spectral seque* *nces are related as n varies. We will not give applications in this paper. However, in work that will ap- pear elsewhere, the second author has used just a small part of the structure in the spectral sequences for computing H*( nX; Z=2) to simplify the proof of some topological nonrealization results of L.Schwartz [Sc]. This structure also appe* *ars to be a reflection of structure in spectral sequences for calculating versions of * *higher Topological Hochschild Homology (see [K3 ]). 1.1. The Smashing Theorem. Our first result is our simplest and most expected. It arises from the natural map between function spaces j : MapT (L, X) ! MapT (K ^ L, K ^ X) that one gets by smashing with the identity map of K. Theorem 1.1. There are natural maps of towers j : P L(X) ! P K^L(K ^ X) with the following properties. (1) There is a commutative diagram of spectra: eL(X) 1 MapT (L, X)+ _________________//_P L(X) ||1 j+ j|| fflffl| eK^L(K^X) fflffl| 1 MapT (K ^ L, K ^ X)+___________//P K^L(K ^ X). (2) The induced map on kth fibers FkL(X) ! FkK^L(K ^ X) is naturally equivalent to the composite Map S(L(k), X^k)h kj-!MapS(K^k ^ L(k), K^k ^ X^k)h k -p*!Map (k) ^k S((K ^ L) , (K ^ X) )h k, 4 AHEARN AND KUHN where p : (K ^ L)(k)! K^k ^ L(k)is the k-equivariant projection. Corollary 1.2. There is a natural map of towers j : P Sm(X) ! P Sm+n( nX) under 1 j+ : 1 ( m X)+ ! 1 ( m+n nX)+ , such that the associated map on kth fibers is equivalent to the map C(m, k)+ ^ k ( -m X)^k ! C(m + n, k)+ ^ k ( -m X)^k induced by the k-equivariant inclusion C(m, k) ,! C(m + n, k). We have listed this theorem and corollary first because it allows us to extend the definition of our towers for 1 nX, with X a space, to towers for 1 1 X, with X a spectrum. Let the spaces {Xn}, n 0,1be the spaces in the spectrum X, so that nXn = 1 X for all n. Then define P S (X) to be the hocolimit over n of the maps of towers n Sn+1 Sn+1 P S (Xn) ! P ( Xn) ! P (Xn+1) where the first map is given by the theorem and the second by the spectrum stru* *c- ture maps. Recalling that hocolimn -n 1 Xn is naturally equivalent to X, the maps 1 nXn+ ! P Sn(Xn) induce maps 1 1 1 X+ ! P S (X). We deduce the following. Corollary 1.3. The kth fiber of the tower P S1(X) is naturally equivalent to the kth extended power C(1, k)+ ^ k X^k ' (X^k)h k. If X1is a 0-connected spectrum, then the connectivity of the maps 1 1 X+ ! PkS (X) increases linearly with k. This identification of both the fibers and convergence of the Goodwillie tower of 1 1 : Spectra! Spectrahas been observed previously by other people, e.g. Goodwillie, Arone, and R.McCarthy (see the comments at the beginning of [McC ])* *.n However, by constructing it in this way, our later1structure theorems for P S (* *X) will immediately imply analogous results about P S (X), and thus results about spectral sequences for computing h*( 1 X). 1.2. The Product and Diagonal Theorems. Our next results are consequences of our study of the map of towers associated to the natural homeomorphisms of function spaces MapT (K _ L, X) = MapT (K, X) x MapT (L, X). To state these, we need to introduce a bit of the language one would use in defining the homotopy category of functors, and also describe an appropriate so* *rt of completed smash product of towers of spectra. For the former, given two functors F and G from pointed spaces to spectra, a weak natural transformation h : F ! G will be a triple (H, f, g), with H a func* *tor from spaces to spectra, g : H ! G a natural transformation, and f : H ! F STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 5 a natural transformation such that f(X) : H(X) ! F (X) is a weak homotopy equivalence for all X. If g(X) is also a weak homotopy equivalence for all X, then we say that h is a weak natural equivalence. Note that, if F and G are homotopy functors, then a weak natural transformation h : F ! G induces a well defined natural transformation in the homotopy category: h(X) = g(X) O f(X)-1 2 [F (X), G(X)]. Furthermore, using homotopy pullbacks, one can define the composition of weak natural transformations. Now we need to define the smash product of two towers of spectra. If P and Q are two towers of spectra, let P ^ Q be the tower with (P ^ Q)k = holimi+jPki^ Qj. Let Fk(P ) denote the homotopy fiber of Pk ! Pk-1. As will be noted in x5.2, there is a weak natural equivalence Y Fk(P ^ Q) ' Fi(P ) ^ Fj(Q). i+j=k Theorem 1.4. There are natural weak homotopy equivalences of towers ~ : P K_L(X) ~-!P K(X) ^ P L(X) with the following properties. (1) There is a commutative diagram of weak natural transformations: eK_L(X) 1 MapT (K _ L, X)+___________________//P K_L(X) || ~| || |o || eK (X)^eL(X) fflffl| 1 MapT (K, X)+ ^ MapT (L, X)+__________//_P K(X) ^ P L(X). (2) The induced weak equivalence on kth fibers Y FkK_L(X) ~-! FiK(X) ^ FjL(X) i+j=k is naturally equivalent to the product, over i + j = k, of the weak natu* *ral transformations Map S((K _ L)(k), X^k)h-kTr-!MapS((K _ L)(k), X^k)h( ix j) -'*!Map (i) (j) ^k S(K ^ L , X )h( ix j) -~ Map (i) ^i (j) ^j S(K , X )h i^ MapS (L , X )h j, where T r is the transfer associated to ix j k, and ' : K(i)^ L(j),! (K _ L)(k)is the ix j-equivariant inclusion. Let r : K _K ! K be the fold map. Since the diagonal map is the composite * MapT (K, X) r--!MapT (K _ K, X) = MapT (K, X) x MapT (K, X), our Product Theorem has consequences for . Let : P K(X) ! P K(X) ^ P K(X) be the weak natural transformation * ~ P K(X) r--!P K_K(X) -!P K(X) ^ P K(X). 6 AHEARN AND KUHN Theorem 1.5. The weak natural transformation has the following properties. (1) There is a commutative diagram of weak natural transformations: eK (X) 1 MapT (K, X)+ ______________________//_P K(X) ||1 + || fflffl| eK (X)^eK (X) fflffl| 1 MapT (K, X)+ ^ MapT (K, X)+ __________//_P K(X) ^ P K(X). (2) The induced weak natural transformation on kth fibers Y FkK(X) ! FiK(X) ^ FjK(X) i+j=k is naturally equivalent to the product, over i + j = k, of the composite* *s of the weak natural transformations MapS (K(k), X^k)h kTr--!MapS(K(k), X^k)h( ix j) ß*-!Map (i) (j) ^k S(K ^ K , X )h( ix j) ~- Map (i) ^i (j) ^j S (K , X )h i^ MapS (K , X )h j, where ß : K(i)^ K(j)! K(k)is the projection. A typical computational consequence of this would be the following. Corollary 1.6. Let h* be a generalized cohomology theory with products. Then the associated spectral sequence for computing h*(MapT (K, X)) is a spectral se- quence of bigraded differential graded h*-algebras. The product on E*,*1corre- sponds to the cup product in h*(MapT (K, X)) in the usual way, and the product E-i,*1 E-j,*1! E-(i+j),*1is induced by the maps on fibers as given in the theor* *em. Specializing to K = Sn, we have some simplification. Corollary 1.7. There is a natural map of towers : P Sn(X) ! P Sn(X)^P Sn(X) under 1 + : 1 ( nX)+ ! 1 ( nX x nX)+ , such that the associated map on kthfibers is equivalent to the product, over i+* *j = k, of the composites C(n, k)+ ^ k ( -n X)^kTr--!C(n, k)+ ^ ix j ( -n X)^k ! (C(n, i) x C(n, j))+ ^ ix j ( -n X)^k, where the second map is induced by the ix j-equivariant inclusion C(n, k) ,! C(n, i) x C(n, j). In this corollary, the second map is an equivalence if n = 1. When n = 1, the (i, j)th component of the map on kth fibers is easily seen to be homotopic to t* *he `shuffle coproduct' X^k ! X^i ^ X^j, STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 7 the sum of the k!=i!j! permutations that preserve the order of the first i and * *last j terms. Note that this induces the usual product on E1 in the classic Eilenber* *g- Moore spectral sequence. 1.3. The Evaluation Theorem. Our next theorem is a consequence of our study of the map of towers associated to the evaluation maps ffl : K ^ MapT (K ^ L, X) ! MapT (L, X). It is convenient to use reduced towers. Let ~P K(X) be the fiber of the proje* *ction P K(X) ! P K(*). Then, for all k, PkK(X) is isomorphic to the product of ~PkK(X) with the sphere spectrum S, and eK (X) induces a natural transformation ~eK(X) : 1 MapT (K, X) ! ~P K(X). Theorem 1.8. There are natural maps of towers ffl : K ^ ~P K^L(X) ! ~P(LX) with the following properties. (1) There is a commutative diagram of spectra: 1K ^~eK^L(X) 1 K ^ MapT (K ^ L, X)__________//_K ^ ~P K^L(X) |1|ffl |ffl| fflffl| ~eL(X) fflffl| 1 MapT (L, X)_________________//_~P(LX). (2) The induced map on kth fibers is naturally equivalent to the composite * K ^ MapS ((K ^ L)(k), X^k)hdk-!K ^ MapS (K ^ L(k), X^k)h k ffl-!Map (k) ^k S(L , X )h k, where the first map is induced by the k-equivariant map of spaces d : K ^ L(k)! (K ^ L)(k) which arises by embedding K diagonally in K^k. In the 1982 paper [K1 ], which studied how the Snaith stable decomposition of n nY interacted with evaluation maps, the second author made use of certain Thom-Pontryagin collapse maps essentially introduced in [M ]. These are explicit k-equivariant maps of spaces fi(m, n, k) : Sm ^ C(m + n, k)+ ! Smk ^ C(n, k)+ . Corollary 1.9. There is a natural map of towers ffl : m ~P Sm+n(X) ! P~Sn(X) under the evaluation 1 ffl+ : 1 ( m m+n X)+ ! 1 ( nX)+ , such that the as- sociated map on kth fibers is equivalent to the map Sm ^ C(m + n, k)+ ^ k ( -m-n X)^k ! C(n, k)+ ^ k ( -n X)^k induced by fi(m, n, k). We note that the effect in mod p homology of this map on fibers is known, so this theorem can be used computationally. 8 AHEARN AND KUHN 1.4. The C(n) operad stucture on P Sn(X). Our final theorem shows that the little n-cubes operad action on nX induces an action on our towers in the expe* *cted way. Recall [M ] that this action is given by suitably compatible maps `(r) : C(n, r) x r ( nX)r ! nX. W Note that ( nX)r = MapT ( rSn, X). We have the following theorem, which will be made more precise in x8. Theorem 1.10. For all n and r, there is a natural map of towers W n n `(r) : C(n, r)+ ^ r P rS (X) ! P S (X) with the following properties. (1) There is a commutative diagram of spectra: 1^eWrSn(X) W n 1 ((C(n, r) x r ( nX)r)+_)________//C(n, r)+ ^ r P rS (X) |1|`(r)+ |`(r)| fflffl| eSn(X) fflffl|n 1 ( nX)+ ________________________//P S (X). (2) The associated map on kth fibers is induced by the operad structure maps C(n, r) x C(n, k1) x . .x.C(n, kr) ! C(n, k), with k1 + . .+.kr = k. Computationally, this implies that the associated spectral sequences for comp* *ut- ing mod p homology admit Dyer-Lashof operations.2 1.5. Organization of the paper. The organization of the paper is as follows. In section 2, we discuss the categories of spectra we work in, and various `nai* *ve' constructions including versions of transfer and norm maps. In section 3, we re* *call the construction of the Arone tower for 1 MapT (K, X)+ , and its homotopical analysis. We use this in section 4 to prove our Smashing and Evaluation Theorem* *s. The Product and Diagonal Theorems are proved section 5, after a brief analysis of the smash product of towers. In section 6 we describe the compatibility among the various transformations of towers defined in ournmain theorems. In section * *7, we deduce our various corollaries for the towers P S , using our explicit equiv* *ariant S-duality maps. Using related constructions with little cubes, Theorem 1.10 is proved in section 8, and, in an appendix, a simplified proof of Arone's converg* *ence theorem is given in the case when K = Sn. This paper includes results from the first author's Ph.D. thesis [Ah ]. The a* *u- thors wish to thank Greg Arone, Bill Dwyer, and Gaunce Lewis for enlightening mathematical discussions on aspects of this project. ____________ 2Exactly what this statement means is still a matter of investigation by the* * authors. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 9 2.Background material on spectra Here we define and discuss various general constructions with spectra that we will later need. By introducing a small amount of fussiness concerning differe* *nt universes, all constructions are of a `naive' nature. The material is essentia* *lly background, and certainly variations of everything we prove here are already kn* *own. 2.1. Spectra and universes. Firstly, we need to specify what we mean by spectra. We find it easiest to work with coordinate free spectra (as in the first pages * *of [LMMS ]). We briefly review the definitions that we need. Let T denote the category of compactly generated based spaces. Fixing an infinite dimensional real inner product space U, one defines an associated cate* *gory of spectra SU. An object X 2 SU assigns a space X(V ) to every finite dimensional subspace V U, and assigns a structure map X(V ) ! W-V X(W ) to every inclusion V W . Here W - V is the orthoganal complement of V in W , and U K = MapT (SU , K) where SU is the one point compactification of U. The structure maps are required to be homeomorphisms. A map of spectra f : X ! Y is a collection of maps f(V ) : X(V ) ! Y (V ) com- patible with the structure maps in the usual way. This makes SU into a topologi* *cal category. If one deletes the requirement that the structure maps be homeomorphisms, one obtains the category of prespectra PSU, and there is a `spectrification' functor l : PSU ! SU, left adjoint to the inclusion of SU in PSU. The category SU has limits and colimits, with limits being formed in PSU, and colimits being formed by applying l to the colimit in PSU. When the universe U is understood and the meaning is clear, we will abbreviate SU to S. With the elementary constructions to be reviewed later in this section, one d* *oes homotopy in the usual way. The stable category hSU is then the category ob- tained from SU by inverting the weak homotopy equivalences. A key observation in this approach to spectra is that any linear isometry U ! U0 induces the same equivalence hSU ' hSU0 on passage to homotopy. We note also that these canon- ical equivalences are compatible with the various constructions given below. S* *ee [LMMS , chapter II] for more detail. 2.2. Suspension spectra. There is an adjoint pair -1 T -! SU 1 defined by 1 X = X(0) and ( 1 K)(V ) = colimW W W+V K. Here U K de- notes SU ^ K, as usual. We let Q = 1 1 : T ! T , and S = 1 S0. When it is necessary to remember U, we will use the notation 1U, etc. (This follows our general rule with all constructions involving spectra: we will be n* *ota- tionally pedantic when it seems prudent.) 2.3. Stablization, elementary smash products and function spectra. Given K 2 T and X 2 SU, we define spectra K ^ X and Map SU(K, X) in SU as fol- lows: K ^ X is the spectrification of the prespectrum with V thspace K ^ X(V ), and Map SU(K, X)(V ) = MapT (K, X(V )). These construction are adjoint to each 10 AHEARN AND KUHN other, Hom SU(K ^ X, Y ) = Hom SU(X, MapSU (K, Y )), and one can deduce various useful isomorphisms in SU [LMMS , p.17, p.20]: MapSU (K ^ L, X) = Map SU(K, MapSU (L, X)), (K ^ L) ^ X = K ^ (L ^ X), and K ^ 1 L = 1 (K ^ L). When clear from context, we will write Map SU(K, X) as MapS (K, X), and then MapS (K, 1 L) as MapS (K, L). A `stabilization' map s : 1 MapT (K, L) ! MapS (K, L) can now be defined as the adjoint to MapT (K, L) MapT(K,j)-------!MapT(K, QL) = 1 MapS (K, L) where j : L ! QL is adjoint to the identity on 1 L. Also arising from adjuncti* *ons are evaluation maps fflT : K ^ MapT (K, L) ! L and fflS : K ^ MapS (K, X) ! X. The next lemma is proved with formal categorical arguments. Lemma 2.1. For any spaces K and X, there is a commutative diagram K ^ 1 MapT (K, X) _1^s_//_K ^ MapS (K, X) || | || |fflS || 1 fflT fflffl| 1 (K ^ MapT (K, X))_________//_ 1 X. A variation on these constructions goes as follows. (Compare with [LMMS , pp.68,69].) Given two universes U and U0, there is an external smash product ^ : SU x SU0 ! S(U U0) defined by letting X ^ Y be the spectrification of the prespectrum with (V W * *)th space X(V ) ^ Y (W ).3 Dually, there is an external mapping spectrum functor: Map SU: SU x S(U U0) ! SU0 defined by Map SU(X, Z)(W ) = Hom SU(X, ZW ), where ZW (V ) = Z(V W ). Again these constructions are adjoint: Hom S(U U0)(X ^ Y, Z) = Hom SU(X, MapSU0(Y, Z)). Again one can formally deduce useful properties, e.g. there are isomorphisms * *in S(U U0): 1UK ^ 1U0L = 1U U0(K ^ L). ____________ 3This formula suffices because subspaces of U U0 of the form V W are cof* *inal among all finite dimensional subspaces. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 11 Another useful property, which follows by a check of the definitions, is that* *, for all K 2 T and Z 2 S(U U0), there are isomorphisms in S(U0): Map SU( 1UK, Z) = Map SU0(K, i*Z), where i : U0 ! U U0 is the inclusion. Given X 2 SU and Z 2 S(U U0), there is an evaluation map ffl : X ^ MapSU (X, Z) ! Z. Precomposing this with ffl : K ^ Map SU(K, X) ! X and then adjointing defines a composition map O : Map SU(K, X) ^ MapSU (X, Z) ! Map S(U U0)(K, Z) for all K 2 T . We will use this construction when defining norm maps in x2.5 below. Given spaces K, L, and spectra X 2 SU, Y 2 SU0, we define ^ : Map SU(K, X) ^ MapSU0(L, Y ) ! Map S(U U0)(K ^ L, X ^ Y ) to be adjoint to the composite of the natural isomorphism (K^L)^Map SU(K, X)^Map SU0(L, Y ) = (K^Map SU(K, X))^(L^Map SU0(L, Y )) with eS ^ eS: (K ^ MapSU (K, X)) ^ (L ^ MapSU0(L, Y )) ! X ^ Y. This is analogous to the usual pairing between mapping spaces ^ : MapT (K, X) ^ MapT (L, Y ) ! MapT (K ^ L, X ^ Y ), and the next lemma records that these constructions are compatible under stabi- lization. Lemma 2.2. For any spaces K, L, X, Y , and universes U, U0, there is a commu- tative diagram in S(U U0) 1UMapT (K, X) ^ 1U0MapT(L, Y )______ 1U U0(MapT (K, X) ^ MapT (L, Y )) |s^s| ||1 ^ fflffl| fflffl| MapSU (K, 1UX) ^ MapSU0(L, 1U0Y ) 1U U0MapT (K ^ L, X ^ Y ) |^| |s| fflffl| fflffl| Map S(U U0)(K ^ L, 1UX ^ 1U0Y_)_____MapS(U U0)(K ^ L, 1U U0(X ^ Y )) Once again, this is proved with formal categorical arguments. The next lemma is standard. Lemma 2.3. If K and L are finite CW complexes, then ^ : Map SU(K, X) ^ MapSU0(L, Y ) ! Map S(U U0)(K ^ L, X ^ Y ) is a weak homotopy equivalence. 12 AHEARN AND KUHN 2.4. Spaces and spectra of natural transformations. If J is a small category, and K : J ! T and X : J ! S are two functors of the same variance, we will write Map JS(K, X) for the spectrum constructed as the categorical equalizer in* * S of the two evident maps Y -! Y MapS (K(j), X(j)) -! MapS (K(j0), X(j00)). j2Ob(J ) ff:j0!j002Mor(J ) Similarly, if K, X : J ! T are two functors of the same variance, one gets a * *space Map JT(K, X), which can be interpreted as the space of natural transformations * *from K to X. The stable and unstable constructions are related by Map JS(K, X)(V ) = Map JT(K, X(V )), for any V 2 U. It is useful to observe that if J = G, a finite group viewed as a category wi* *th one object, then Map GS(K, X) is the categorical fixed point spectrum of the na* *ive G-spectrum MapS (K, X) with conjugation G-action. In this case, we also write X=G for the categorical orbit spectrum. 2.5. Norm maps, transfers, and Adams isomorphisms. In this subsection, we give quick definitions of transfer and norm maps suitable for our later homo- topical identification of natural transformations between fibers in the Arone t* *owers. These definitions are adapted to our setting, but are intended to agree in the * *homo- topy category with anyone else's transfer and norm maps. As far as the authors * *can tell, constructions of norm maps using only än ive" constructions first appeare* *d in the literature in the 1989 paper of Weiss and Williams [WW , x2]. (Those autho* *rs credit Dwyer with some of these ideas, and, of course, Adams' paper [Ad ] was i* *n- fluential.) Our definitions are small perturbations of those in the recent prep* *rint of John Klein [Kl]. Proposition 2.9, which relates transfer and norm maps, appears to be new in the literature, and a desire for a transparent proof of this has g* *uided our constructions. Let G be a finite group, and call a spectrum with G-action a G-spectrum. Fix two universes U and U0, and let i : U0 ! U U0 be the inclusion. Definitions 2.4. Given a subgroup H G, and a G-spectrum X 2 S(U U0), we define the homotopy fixed point and homotopy orbit spectra as follows. (1) XhH = Map HS(U U0)(EG+ , X). (2) XhH = (EG+ ^ MapHSU(EG+ , 1UG+ ) ^ i*X)=G. The first definition is, we trust, expected. The corollary of the next lemma * *says that the second has the correct homotopy type. Lemma 2.5. There is a weak equivalence of G-spectra in SU 1 G=H+ ' Map HS(EG+ , G+ ). Proof.There are weak equivalences and isomorphisms of G-spectra: 1 G=H+ -~!MapS (G=H+ , S) = Map HS(G+ , S) -~!Map H S (EG+ ^ G+ , S) = Map HS(EG+ , MapS (G+ , S)) -~ Map H S (EG+ , G+ ). STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 13 Here the first and last maps arise in the same manner. If K < G is any subgroup, there is a commutative diagram of G-spectra W 1 gK2G=K S _________ G=K+ '|| || Q fflffl| |fflffl gK2G=K S ______MapS(G=K+ , S) where the left vertical map is the inclusion of the wedge into the product, a w* *eak homotopy equivalence. Corollary 2.6. There is a weak natural equivalence in S(U U0) XhH ' (EG+ ^ X)=H. Proof.There are weak natural equivalences XhH ' (EG+ ^ 1UG=H+ ^ i*X)=G ~-!(EG+ ^ G=H+ ^ X)=G = (EG+ ^ X)=H. Here the first equivalence is a consequence of the lemma, and the second follows from the fact that, very generally, there is a natural weak equivalence 1UK^i** *X ! K ^ X. Our transfer maps are defined as follows. Definitions 2.7. Let K H G, and let X be a G-spectrum in S(U U0). (1) Let trHK: Map HSU(EG+ , G+ ) ! Map KSU(EG+ , G+ ) be the inclusion of fi* *xed point spectra. (2) Let T rHK(X) : XhH ! XhK be the natural map induced by trHK. We sketch a proof that T rHK(X), viewed as a natural transformation of functo* *rs on the homotopy category of spectra with G-action, agrees with other standard constructions of the transfer, in particular, the transfer arising from [LMMS * *]. Both of these transfers behave well with respect to pushouts and weak equivalences i* *n the X variable, and with respect to forgetful functors arising from subgroup inclus* *ions. Using these facts one can reduce to just needing to show that the two definitio* *ns of T rGH(G+ ) agree up to weak equivariant homotopy. For us, this map is equivalent to the map S ~-!MapS (G=G+ , S) ! MapS (G=H+ , S) induced by the projection ß : G=H+ ! G=G+ . Now one checks that this agrees with the composite S ø-! 1 G=H+ -~!MapS (G+ , S) where ø is the pretransfer of [LMMS , p.181]. We now define our norm maps. Definition 2.8. Given H G, and a G-spectrum X in S(U U0), let H (X) : XhH ! XhH be defined as follows. First note that i*X = Map GSU0(G+ , i*X) = Map GSU( 1UG+ , X). 14 AHEARN AND KUHN Now consider composition O : Map SU(EG+ , 1UG+ ) ^ MapGSU( 1UG+ , X) ! Map S(U U0)(EG+ , X). This is G-equivariant with respect to the usual conjugation G-action on the two terms Map SU(EG+ , 1UG+ ) and Map S(U U0)(EG+ , X). Taking H fixed points then yields a map of spectra Map HSU(EG+ , 1SUG+ ) ^ i*X ! XhH . Now one notes that this map is invariant with respect to the diagonal G-action * *on the domain, where G acts on (Map HSU(EG+ , 1SUG+ ) by acting on the right on G* *+ . Thus one has an induced map (Map HSU(EG+ , 1SUG+ ) ^ i*X)=G ! XhH . H (X) is then obtained by precomposing this map with the map XhH ! (Map HSU(EG+ , 1SUG+ ) ^ i*X)=G induced by EG+ ! S0. By construction, the following proposition is self evident. Proposition 2.9. Given K H G, and X 2 S(U U0), there is a commutative diagram of spectra H(X) XhH _____//XhH TrHK(X)|||| fflffl| Kfflffl|(X) XhK _____//XhK where the unlabelled vertical arrow is the inclusion of fixed point spectra. Norm maps should be equivalences under suitable freeness and finiteness con- ditions. In homotopy, such equivalences have been terms `Adams isomorphisms' [LMMS ]. The version we need goes as follows. Proposition 2.10. If X = MapS (K, Y ), where Y is any G-spectrum and K is any finite free G-CW complex, then the norm map H (X) : XhH ! XhH is a weak homotopy equivalence for all H < G. Proof.We show that H (X) is an equivalence in various cases. When X = 1U U0G+ , via the weak equivalences of Lemma 2.5, H (X) corresponds to the isomorphism ( 1UG=H+ ^ 1U0G+ )=G = 1U U0G=H+ . Now we note that both the domain and range of H (X) preserve equivariant weak homotopy equivalences and homotopy cofiber sequences. Thus, by induction on cells, H (X) is an equivalence if Y is any G-spectrum equivalent to a finite G* *-CW spectrum, and K is any finite free G-CW complex. Now we note that, if K is a finite free G-CW complex, then both the domain and range of H (MapS (K, Y )) commute with homotopy colimits in the Y variable. The proposition follows. An application of this that we will need later goes as follows. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 15 Corollary 2.11. Let G and H be two finite groups. If X is a G-spectrum, K a finite free G-CW complex, Y an H-spectrum, and L a finite free H-CW spectrum, then ^ : Map GSU(K, X) ^ MapHSU0(L, Y ) ! Map GxHS(U(U0)K ^ L, X ^ Y ) is a weak homotopy equivalence. Proof.Let A = Map S(K, X), B = Map S(L, Y ), and C = Map S(K ^L, X ^Y ). We wish to show that the G x H-equivariant weak equivalence A ^ B ! C induces an equivalence AG ^ BH ! CGxH . Since there are equivalences AG -~!AhG -~ AhG, and similarly for B and C, it suffices to show that AhG ^ BhH ! Ch(GxH) is an equivalence. But this is clear, as easy formal arguments show that AhG ^ BhH = (A ^ B)h(GxH). Finally we note Corollary 2.12. Let G be a finite group. If K is a finite free G-CW complex, th* *en for all G-spectra Y 2 S(U U0), there is a weak natural equivalence (Map SU(K, S) ^ i*Y )hG ' Map GS(U U0)(K, Y ) Proof.This arises from the equivalences (MapS (K, S) ^ Y )hG ~-!MapS (K, Y )hG -~ Map GS(K, Y ). 3.The Arone model In this section we review the Greg Arone's explicit construction of the tower P K(X) of fibrations of spectra under 1 MapT (K, X)+ , tweaked a bit to lend itself to the homotopical analysis we are interested in.4 3.1. Definition of the tower. We introduce a small category central to our work. Definition 3.1. Let E denote the category with objects the finite sets 0 = ; and n = {1, 2, . .,.n}, n 1, and morphisms the surjective functions. This has fu* *ll subcategories ~Ewhose objects are n with n 1, Ek, whose objects are n with n k, and ~Ek= ~E\ Ek.5 Fundamental E-spaces are the following. Definition 3.2. If X is a pointed space, let X^ : Eop ! T be the following func* *tor. On objects, it assigns to n, X^n, the n-fold smash product of X with itself, wh* *ere we use the convention that X^0 = S0. On morphisms, it assigns to a surjective function ff : n ! m, the associated `diagonal' map ff* : X^m ! X^n sending x1^ . .^.xm to xff(1)^ . .^.xff(n). Note that this is well defined precisely be* *cause ff is surjective. ____________ 4For example, Arone works with the functors from (based) spaces to spaces wh* *ich send a space X to Q MapT(K, X). It seems more natural to regard the basic functors as going * *from spaces to a suitable category of spectra. 5This category E of epimorphisms was called M by Arone in [Ar] and Mopby McC* *arthy in [McC ]. The senior author objects. 16 AHEARN AND KUHN Armed with these E-spaces, we can define Arone's towers in our setting. Definitions 3.3. Given two spaces K and X in T , define spectra P1K(X), PkK(X), ~P1K(X), and ~PkK(X), by the formulae P1K(X) = Map ES(K^ , X^ ), PkK(X) = Map EkS(K^ , X^ ), P~K1(X) = Map ~ES(K^ , X^ ), ~PkK(X) = Map ~EkS(K^ , X^ ). There are evident restriction maps pk : PkK(X) ! PkK-1(X), defining a tower P K(X), compatible with restrictions maps qk : P1K(X) ! PkK(X), and P1K(X) = limkPkK(X). As we will discuss below, the maps pk are fibrations in S; thus P1K(X) is also equivalent to the homotopy limit of the tower. Notation 3.4. Let FkK(X) be the fiber of pk : PkK(X) ! PkK-1(X). The reduced functors are related to the unreduced functors by PkK(X) = ~PkK(X) x S, (with the product in S), and they similarly form a tower with limit (and holimi* *t) ~P1K(X). Now we define natural transformations 1 MapT (K, X)+ ! P K(X). For n 0, let eK (X, n) : 1 MapT (K, X)+ ! Map S(K^n, X^n) be the composite 1 ,n s 1 MapT (K, X)+ ----! 1 MapT (K^n, X^n) -!MapS (K^n, X^n), where ,n : MapT (K, X)+ ! MapT (K^n, X^n) sends a function f to f^n . (When n = 0, this means the evident projection MapT (K, X)+ ! S0.) Lemma 3.5. Let ff : n ! m be a surjective function. Then, for all K and X there is a commutative diagram in S eK (X,n) 1 MapT (K, X)+ _____//_MapS(K^n, X^n) eK|(X,m)| |ff*| fflffl| ff* fflffl| MapS (K^m , X^m )____//MapS(K^m , X^n). STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 17 Proof.The analogous diagram in T , MapT (K, X)+___,n__//_MapT(K^n, X^n) ,m|| |ff*| fflffl| ff* fflffl| MapT (K^m , X^m )____//MapT(K^m , X^n), clearly commutes, and the lemma follows, using the naturality of s. This lemma says that the maps Y Y eK (X, n) : 1 MapT (K, X)+ ! MapS (K^n, X^n) n n are equalized by the maps defining P1K(X) = Map ES(K^ , X^ ). Thus they define eK1(X) : 1 MapT (K, X)+ ! P1K(X), and then eKk(X) = qk O eK1(X) : 1 MapT (K, X)+ ! PkK(X). The main theorem of [Ar] is a convergence result. In our context, it reads as follows. Theorem 3.6. Let K be a finite CW complex and X a space with connectivity at least as large as the dimension of K. Then eKk(X) is (1+conn X-dim K)(1+k)-1 connected. In particular, eK1(X) is a weak homotopy equivalence and the tower is strongly convergent. In Appendix A, we will outline how the proof of this theorem goes in the case when K is a sphere. Needed first in any proof, however, is an analysis of the homotopical behavior of the tower P*K(X). We now proceed to make this analysis, as these results are also needed to prove our main theorems. 3.2. Homotopical analysis of the tower. The key to understanding P K(X) homotopically is to take advantage of two observations. Firstly, the filtration* * of E by the subcategories Ek induces a natural filtration on contravariant functors * *from E to T or S. Secondly, this filtration on the particular functors K^ is particu* *larly nice, and essentially exhibits them as cofibrant Eop-objects in the functor cat* *egories. We should say immediately that these sorts of observations have been made before. See, for example, [McC , Appendix A], [L, x3], as well as Arone's own paper [Ar]. These are all modern references, but, slightly disguised, these ide* *as are certainly much older. With C either T or S, let CE denote the category of contravariant functors X : Eop ! C. The inclusion ik : Ek ,! E induces the restriction i*k: CE ! CEk w* *ith left adjoint ik*: CEk ! CE, and we let Xk = ik*i*kX. Formally, one sees that, f* *or all X 2 T Eand Y 2 CE, there are natural isomorphisms in C Map EC(Xk, Y ) = Map EkC(X, Y ). Explicitly, Xk(n) = colimn#E~X, k 18 AHEARN AND KUHN where n # Ek denotes the usual category under n with objects n ! j in E with j k, and ~X(n ! j) = X(j). The Xk assemble into a filtration of X, X0 ! X1 ! X2 ! . .,. and, noting that Xk(n) = X(n) for all k n, one sees that X is realized as the colimit. In the next lemma, k denotes the symmetric group on k letters, viewed as the morphisms k ! k in E. Lemma 3.7. For all X 2 T E, there is a pushout in T E E( , k)+ ^ k Xk-1(k)____//Xk-1 | | | | fflffl| fflffl| E( , k)+ ^ k Xk(k)_____//_Xk. Proof.This follows from the observation that there is a pushout diagram of small categories E(n, k) x k (k # Ek-1)__//(n # Ek-1) | | | | fflffl| fflffl| E(n, k) x k (k # Ek)____//(n # Ek). Corollary 3.8. For all X 2 T Eand Y 2 CE there is a pullback diagram in C Map EC(Xk, Y_)_____//MapCk(Xk(k), Y (k)) | | | | fflffl| fflffl| Map EC(Xk-1, Y_)___//MapCk(Xk-1(k), Y (k)). Now suppose that X = K^ . Observe that K^k(k) = K^k, and that K^k-1(k) is the fat diagonal k(K) inside K^k. If K is a CW complex, then K^k can be ob- tained from k(K) by attaching only free k-cells. The fact that (K^k, k(K)) is an equivariant CW pair implies that the inclusion k(K) ! K^k is an equivariant cofibration. Recalling that K(k)denotes K^k= k(K) (as in [Ar]), we conclude Proposition 3.9. Map Sk(K^k, X^k) ! Map Sk( k(K), X^k) is a fibration with fiber Map Sk(K(k), X^k). Thus pk : PkK(X) ! PkK-1(X) is a fibration with fiber FkK(X) = Map Sk(K(k), X^k). Furthermore, there are natural weak equivalences FkK(X) ~-!Map S(K(k), X^k)h k ~- (Map S(K(k), S) ^ X^k)h k. The last statement here makes it clear that the Arone tower has the form of a Goodwillie tower. In the language of [G3 ], we conclude STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 19 Corollary 3.10. The kth Taylor coefficient of the functor sending a space X to the spectrum 1 MapT (K, X) is the k-spectrum MapS (K(k), S). 4. The Smashing and Evaluation Theorems 4.1. The Smashing Theorem. The first of our general theorems studying addi- tional structure in the Arone tower involves smashing with a constant space K. There is an unstable map j : MapT (L, X) ! MapT (K ^ L, K ^ X) which sends a function f : L ! X to 1K ^ f : K ^ L ! K ^ X. Note that this map can be written as the composite MapT (L, X) ! MapT (K ^ L, K ^ L ^ MapT (L, X)) ! MapT (K ^ L, K ^ X) where the first map is a unit of an adjunction, and the second map is induced by fflT : L ^ MapT (L, X) ! X. Replacing T by S in this composite then similarly defines j : MapS (L, X) ! MapS (K ^ L, K ^ X), and the unstable and stable maps will be compatible under stabilization in the evident way. Theorem 4.1. There are natural maps of towers j : P L(X) ! P K^L(K ^ X) with the following properties. (1) There is a commutative diagram in S: eL(X) 1 MapT (L, X)+ _________________//_P L(X) |j| j|| fflffl| eK^L(K^X) fflffl| 1 MapT (K ^ L, K ^ X)+___________//P K^L(K ^ X). (2) The induced map on kth fibers is the composite MapSk(L(k), X^k)j-!MapSk(K^k ^ L(k), K^k ^ X^k) -p*!Map k (k) ^k S ((K ^ L) , (K ^ X) ), where p : (K ^ L)(k)! K^k ^ L(k)is the k-equivariant projection. Corollary 4.2. The natural transformation of functors of X, 1 j : 1 MapT (L, X) ! 1 MapT (K ^ L, K ^ X) induces, on kth Taylor coefficients, the k-equivariant map of spectra * MapS(L(k), S) j-!MapS(K^k ^ L(k), K^k) p-!MapS ((K ^ L)(k), K^k). Proof of Theorem 4.1.Given a surjection ff : n ! m, we define jff: MapS (L^m , X^n) ! MapS ((K ^ L)^m , (K ^ X)^n) to be the composite MapS (L^m , X^n)j-!MapS(K^m ^ L^m , K^m ^ X^n) ff*--!Map ^m ^m ^n ^n (SK ^ L , K ^ X ). 20 AHEARN AND KUHN Note that jffis natural in all variables. Given surjections n fi-!m ff-!l, one easily verifies that there is a commutat* *ive diagram jfi (4.1) MapS (L^m , X^n)____//_MapS((K ^ L)^m , (K ^ X)^n) ff*|| ff*|| fflffl| jffOfi fflffl| MapS (L^l,OX^n)_____//MapS((KO^ L)^l,O(KO^ X)^n) fi*|| fi*|| | jff | MapS (L^l, X^m )____//_MapS((K ^ L)^l, (K ^ X)^m ). Recall that P1L(X) is defined as an equalizer. A first consequence of (4.1) i* *s that there is a commutative diagram Y _______________//_Y ^l ^m MapS (L^n, X^n)_______________//_ MapS(L , X ) n m-ff!l |Q| |Q | nj | ffjff Y fflffl| _____//Y |fflffl MapS ((K ^ L)^n, (K ^ X)^n)____// MapS ((K ^ L)^l, (K ^ X)^m ). n m-ff!l By taking equalizers, this then induces a filtration preserving natural map j : P1L(X) ! P1K^L(K ^ X), and thus a map on the associated towers. To identify the induced map on fibers, we first note that jff: MapS (L^m , X^n) ! MapS ((K ^ L)^m , (K ^ X)^n) also equals the composite MapS (L^m , X^n)j-!MapS(K^n ^ L^m , K^n ^ X^n) ff*--!Map ^m ^m ^n ^n (SK ^ L , K ^ X ). This makes it clear that the k-equivariant map lim jff: MapS ( k(L), X^k) ! MapS ( k(K ^ L), (K ^ X)^k) ff2(k#Ek-1) can be identified as the composite MapS ( k(L), X^k)j-!MapS(K^k ^ k(L), K^k ^ X^k) ! Map S( k(K ^ L), (K ^ X)^k), and the last part of the theorem follows. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 21 Finally, statement (1) of the theorem is a consequence of the following commu- tative diagram: 1 MapT (L, X) ____j_____//_ 1 MapT (K ^ L, K ^ X) ||1 ,k ||1 ,k fflffl| j fflffl| 1 MapT (L^k, X^k) ____//_ 1 MapT ((K ^ L)^k, (K ^ X)^k) |s| |s| fflffl| j fflffl| MapS (L^k, X^k)________//MapS((K ^ L)^k, (K ^ X)^k). 4.2. The evaluation theorem. The next of our general theorems concerns the compatiblity of the Arone tower with generalized evaluation maps. We have maps ffl : K ^ MapT (K ^ L, X) ! MapT (L, X) defined as the adjoint to the evaluation map fflT : K ^ L ^ MapT (K ^ L, X) ! X already discussed in x2.3. Using the stable evaluation fflS, we similarly have maps ffl : K ^ MapS (K ^ L, X) ! MapS (L, X). These unstable and stable generalized evaluation maps are compatible under sta- blization, courtesy of Lemma 2.1. Theorem 4.3. There are natural maps of towers ffl : K ^ ~P K^L(X) ! ~P(LX) with the following properties. (1) There is a commutative diagram in S: 1K ^~eK^L(X) 1 K ^ MapT (K ^ L, X)__________//_K ^ ~P K^L(X) ffl|| |ffl| fflffl| ~eL(X) fflffl| 1 MapT (L, X)_________________//_~P(LX). (2) The induced map on kth fibers is the composite * K ^ MapSk ((K ^ L)(k), X^k)d-!K ^ MapSk (K ^ L(k), X^k) ffl-!Map k(k) ^k S (L , X ), where the first map is induced by the k-equivariant map of spaces d : K ^ L(k)! (K ^ L)(k) which arises by embedding K diagonally in K^k. 22 AHEARN AND KUHN Corollary 4.4. The natural transformation of functors of X, 1 ffl : 1 K ^ MapT (K ^ L, X) ! 1 MapT (L, X) induces, on kth Taylor coefficients, the k-equivariant map of spectra * ffl K ^ MapS ((K ^ L)(k), S) d-!K ^ MapS (K ^ L(k), S) -!MapS (L(k), S). Proof of Theorem 4.3.Given a surjection ff : n ! m, we define fflff: K ^ MapS ((K ^ L)^m , X^n) ! MapS (L^m , X^n) to be the composite * K ^ MapS (K^m ^ L^m , X^n)d-!K ^ MapS (K ^ L^m , X^n) ffl-!Map ^m ^n S(L , X ), where the first map is induced by the diagonal d : K ! K^m . Note that fflffis natural in all variables. Given surjections n fi-!m ff-!l, one easily verifies that there is a commutat* *ive diagram fflfi K ^ MapS ((K ^ L)^m , X^n)____//MapS(L^m , X^n) |ff*| ff*|| fflffl| fflffOfi fflffl| K ^ MapS ((KO^OL)^l, X^n)_____//MapS(L^l,OX^n)O |fi*| fi*|| | fflff ^l| ^m K ^ MapS ((K ^ L)^l, X^m )___//_MapS(L , X ). As in the proof of Theorem 4.1, there is thus an induced natural map of towers ffl : K ^ ~P K^L(X) ! ~P(LX). The induced map on fibers is easy to identify. Note that the k-equivariant m* *ap lim fflff: K ^ MapS ( k(K ^ L), X^k) ! MapS ( k(L), X^k) ff2(k#Ek-1) can be identified as the composite K ^ MapS ( k(K ^ L), X^k)! K ^ MapS (K ^ k(L), X^k) ffl-!Map ^k S( k(L), X ), where the first map is induced by the equivariant inclusion K^ k(L) ! k(K^L). The last part of the theorem follows. Statement (1) of the theorem is a consequence of the following commutative diagram: 1 K ^ MapT(KV^VL,X)1^,k//_ 1 K ^ MapT((K ^ L)^k,X^k)1^s//_K ^ MapS((K ^ L)^k,X* *^k) || VVVVVVV^,k | | | VVVV |^1 |^1 | VVVVV** fflffl| 1^s fflffl| |ffl 1 K^k^ MapT((K ^ L)^k,X^k)_//K^k^ MapS((K ^ L)^k,X^k) | | ffl| ffl| | | | fflffl| ,k fflffl| s fflffl| 1 MapT(L,X)_______// 1 MapT((K ^ L)^k,(K ^_X)^k)//_MapS((K ^ L)^k,(K ^ X)^k* *). STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 23 5.The Product and Diagonal Theorems The goal of this section is to prove a general result about how the unstable homeomorphisms MapT (K _ L, X) = MapT (K, X) x MapT (L, X) lead to pairings among the associated Arone towers. One thus gets pairings among the spectral sequences which arise after applying any multiplicative cohomology theory to the towers. In discussing and proving our result, it seems necessary to first say a littl* *e about the general theory of the smash product of towers. The results here are unsurpr* *ising and presumably known, but we have been unable to find them explicitly in the literature. 5.1. Homotopy limits of spectra. If J is a small category, EJ+ : J ! T is defined by letting EJ (j)+ = B(J # j)+ . Given a functor Y : J ! S, its homotopy limit is the spectrum defined by the formula holimJY = Map JS(EJ+ , Y ). This construction is natural with respect to both the functor and the category. We will make use of the following construction. Given X 2 S and Y : J ! S there is a natural map (5.1) X ^ holimJY ! holimJ(X ^ Y ) defined as the adjoint to the composite Map JS(EJ+ , Y ) ! Map JS(X ^ EJ+ , X ^ Y ) = MapS (X, MapJS(EJ+ , X ^ Y )). Similarly, there is a natural map (5.2) (holimJY ) ^ X ! holimJ(Y ^ X) _ Y ! The notation X xZ Y will denote the homotopy pullback holim fflffl|. X __//Z The homotopy fiber of a map Y ! Z is then defined as * xZ Y . 5.2. Smash products of towers of spectra. If we let N denote the poset 0 < 1 < 2 < . .,.then a tower of spectra can be regarded as a functor P : Nop ! S. Given towers P and Q, respectively taking values in SU and SU0, the external smash product yields a bi-tower P ^ Q : (N x N)op ! S(U x U0). When the context is clear, we will repress explicit notation for universes, and thus jus* *t write P ^ Q : (N x N)op! S. Now suppose given a general bi-tower C : (N x N)op ! S. One then has an associated tower (which we still call C) with kth term given by Ck = holimi+jCki,j. This tower can be understood homotopically via the next lemma. 24 AHEARN AND KUHN Lemma 5.1. Given a bi-tower C : (N x N)op! S, the diagram Q Ck _____________//_i+j=kCi,j | | | | fflffl| Q fflffl| Ck-1 ____//_i+j=kCi-1,jxCi-1,j-1Ci,j-1 naturally homotopy commutes, and induces a natural homotopy equivalence on ho- motopy fibers. Sketch proof.Let Ni N be the poset 0 < 1 < . .<.i, and let (N x N)k N x N be the poset {(i, j) | i + j k}. There is a pushout of posets: ` i+j=kNix Nj- {(i, j)}____//(N x N)k-1 | | | | ` fflffl| fflffl| i+j=kNix Nj __________//(N x N)k. This induces a pullback of spectra Q Ck ________//_i+j=kholimNixNjC | | | | fflffl| Q fflffl| Ck-1 ____//_i+j=kholimNixNj-{(i,j)}C in which the two vertical maps are fibrations. Now note that holimNC ! C(i, j) ixNj is an equivalence, as (i, j) is the terminal object in Nix Nj, and holim C ! Ci-1,jxCi-1,j-1Ci,j-1 NixNj-(i,j) 8 9 < (i-1,j)OO = is an equivalence, as : | is cofinal in Nix Nj- {(i, j)}. (i-1,j-1)//_(i,j-1); As in the introduction, if P is a tower, we let Fk(P ) denote the homotopy fi* *ber of Pk ! Pk-1. Similarly, if C is a bi-tower, we let Fi,j(C) denote the homotopy fi* *ber of Ci,j! Ci-1,jxCi-1,j-1Ci,j-1. This is the same thing as the iterated homotopy fiber of the square Ci,j_______//Ci,j-1 | | | | fflffl| fflffl| Ci-1,j_____//Ci-1,j-1. (See [G2 , 1.1b].) Specializing to the case when C = P ^ Q, use of construction (5.2), and then (5.1), thus yields the composite (5.3) Fi(P ) ^ Fj(Q) ! Fi(P ^ Fj(Q)) ! Fi,j(P ^ Q). STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 25 Lemma 5.2. Fi(P )^Fj(Q) ! Fi,j(P ^Q) is a natural weak equivalence of spectra. Sketch proof.As a fibration sequence in spectra is homotopy equivalent to a cof* *i- bration sequence, smashing a fibration sequence with a spectrum yields a sequen* *ce equivalent to a fibration sequence. Applying this principle to each of the two * *maps in (5.3) shows each to be an equivalence. The two lemmas together have the following corollary. Corollary 5.3. If P and Q are towers of spectra, the natural maps Y Y Fk(P ^ Q) ! Fi,j(P ^ Q) Fi(P ) ^ Fj(Q) i+j=k i+j=k are weak equivalences. We now turn to a discussion of spectral sequences. Suppose h is a spectrum. Applying h* and h* to a tower P yields left half plane homology and cohomology spectral sequences {Er*,*(P )} and {E*,*r(P )} with E1-k,*(P ) = h*-k(Fk(P )) and E-k,*1(P ) = h*-k(Fk(P )). Now suppose h is a ring spectrum. Then the last corollary implies that, given towers P and Q, there are natural pairings E1-i,m(P ) E1-j,n(Q) ! E1-(i+j),m+n(P ^ Q) and E-i,m1(P ) E-j,n1(Q) ! E-(i+j),m+n1(P ^ Q). One then has Theorem 5.4. The pairings extend to pairings of spectral sequences. The pairings have the expected properties. For example, in the cohomology spectral sequence, the pairings are related to the pairing of filtered groups colimih*(Pi) colimjh*(Qj) ! colimkh*((P ^ Q)k) at the level of E1 . A careful proof of this theorem seems to not appear in the literature. Howeve* *r, a proof can be constructed in a straightforward manner by carefully mimicking the discussion on pp.660-668 of G.W.Whitehead's book [Wh ] where he discusses pairings in spectral sequences associated to products of filtered spaces. The t* *rans- lation into our setting involves arguments of the sort given in the proofs of o* *ur two lemmas, but no other new ideas. 26 AHEARN AND KUHN 5.3. The Product Theorem. The homeomorphism MapT (K _ L, X)+ = MapT (K, X)+ ^ MapT (L, X)+ suggests that there should be a compatible weak equivalence between the two tow* *ers P K_L(X) and P K(X) ^ P L(X). We will shortly see that this is the case. To be computationally useful, we will need to also identify the induced weak equivalence on fibers. By Corollary 5.3, we have homotopy equivalences Fk(PK (X)^PL(X)) ~-!Y Fi,j(PK (X)^PL(X)) ~- Y MapSi(K(i), X^i)^MapSj(L(i), X^* *j). i+j=k i+j=k Meanwhile, the k-equivariant homeomorphism ` (5.4) k+ ^ ix j (K(i)^ L(j)) = (K _ L)(k) i+j=k induces an isomorphism Y x FkK_L(X) = MapSi j(K(i)^ L(j), X^k) i+j=k To state our main product theorem it is convenient to define a bi-tower P K,L* *(X) by the formula PiK,L,j(X) = Map EixEjS(K^ ^ L^, a*X^ ). Here a*X^ : (E x E)op! T is defined by a*X^ (i, j) = X^(i+j). Theorem 5.5. There are natural homotopy equivalences of towers P K_L(X) ~-!P K,L(X) -~ P K(X) ^ P L(X) with the following properties. (1) There is a commutative diagram in S: eK_L(X) 1 MapT (K _ L, X)+___________________//P K_L(X) || || |o || | || fflffl| || P K,L(X) || OO || | || |o || eK (X)^eL(X) | 1 MapT (K, X)+ ^ MapT (L, X)+__________//_P K(X) ^ P L(X). (2) The induced equivalences on kth fibers FkK_L(X) ~-!Fk(P K,L(X)) -~ Fk(P K(X) ^ P L(X)) STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 27 fits into a commutative diagram of equivalences FK_Lk(X)______~_______//_Fk(PK,L(X))oo_~________Fk(PK (X) ^ PL(X)) || | | || |o |o Y || Y fflffl| Y fflffl| MapSix j(K(i)^ L(j),X^k)~//_ Fi,j(PK,L(X))o~o_____ Fi,j(PK (X) ^ PL(X)) i+j=k i+j=k i+j=k OO || | || |o Y || Y | MapSix j(K(i)^ L(j),X^k)oo_____~____________ MapSi(K(i),X^i) ^ MapSj(L(* *j),X^j), i+j=k i+j=k where the bottom map is the product of the evident smash product maps. We will use the notation ~ : P K_L(X) ~-!P K(X) ^ P L(X) to denote the weak natural equivalence of the theorem. Remark 5.6. In stating this theorem, we have continued to repress notation for universes. However, we hope it is understood that, if P K(X) is a tower in SU, * *and P L(X) is a tower in SU0, then the maps and objects in the theorem are living in S(U U0). The key to our product theorem is the following observation, hinted at in (5.* *4) above. Let a : E x E ! E be the functor defined by a(i, j) = (i + j). This indu* *ces a functor a* : T E! T ExE by pullback. This has a left adjoint a* : T ExE! T Eexplicitly given by a*X(k) = colimk#ExEX where k # E xE is the category with objects all triples (i, j, ff) where ff : k* * ! (i + j) is a surjection. Then one has Lemma 5.7. The inclusions K^m ^ L^n (K _ L)^m+n induce an isomorphism in T E, a*(K^ ^ L^) = (K _ L)^, and this restricts to give isomorphisms for all k a*(colimi+j(kK^i^ L^j)) = (K _ L)^k. We also record the following fact. Lemma 5.8. hocolimi+j(kK^i^ L^j) ! colimi+j(kK^i^ L^j) is an equivalence in T E* *xE. Proof.Let (N x N)k denote the full subcategory of N x N with objects all (i, j) such that i + j k. It is easy to see that, for all n, K, and L, the functor * *on N x N sending (i, j) to the space K^i(n) ^ L^j(n) is a cofibrant object in the * *model category structure on T (NxN)kdescribed in [DS , x10.13]. The lemma follows. The maps in the theorem are now easy to define. A natural equivalence PkK_L(X) ! PkK,L(X) 28 AHEARN AND KUHN is defined by PkK_L(X) = Map ES((K _ L)^k, X^ ) = Map ES(a*(colimi+j(kK^i^ L^j)), X^ ) = Map ExES(colimi+j(kK^i^ L^j), a*X^ ) ~-!MapExE ^ ^ * ^ S (hocolimi+j(kKi ^ Lj), a X ) = holimi+jMkapExES(K^i^ L^j, a*X^ ) = holimi+jPkK,Li,j(X) = PkK,L(X). The map of towers P K(X) ^ P L(X) ! P K,L(X) is even more evident. It is the map on towers induced by the map of bi-towers ^ : Map EiS(K^ , X^ ) ^ MapEjS(L^, X^ ) ! Map EixEjS(K^ ^ L^, a*X^ ). (This is not so evidently a homotopy equivalence, but, we will learn that it is* *.) The theorem is now easily proved. First, we check that the diagram in (1) commutes. This will follow if we veri* *fy that, for all i + j = k, the diagram K_L(X,k) 1 MapT (K _ L, X)+______e____________//_MapS((K _ L)^k, X^k) || || | || | || fflffl| || MapS(K^i^ L^j, X^k) || OO || | || | || eK (X,i)^eL(X,j) | 1 MapT (K, X)+ ^ MapT(L, X)+_________//_MapS(K^i, X^i) ^ MapS(L^j, X^j) commutes. But this diagram commutes, as one easily checks that the diagram of spaces MapT (K _ L, X)________,k________//_MapT((K _ L)^k, X^k) || || | || | || fflffl| || MapT (K^i^ L^j, X^k) || OO || | || | || ,ix,j | MapT(K, X) x MapT(L, X)_________//_MapT(K^i, X^i) x MapT(L^j, X^j) commutes, and then Lemma 2.2 implies that the diagram STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 29 1 MapT((K _ L)^k, X^k)_______s_________//MapS((K _ L)^k, X^k) | | | | fflffl| s fflffl| 1 MapT(K^i^OL^j,OX^k)___________________//MapS(K^i^OL^j,OX^k) | | | | | | 1 MapT(K^i, X^i) ^ MapT(L^j, X^j)s^s___//_MapS(K^i, X^i) ^ MapS(L^j, X^j) commutes. The lower rectangle in (2) commutes by inspection, the top left square commut* *es by definition, and the top right square commutes by naturality. In this diagram* *, the two downward arrows are equivalences arising from Lemma 5.1, the top rightward arrow is an equivalence as it is induced by an equivalence of towers, and the l* *ower leftward arrow is an equivalence by Corollary 2.11. It follows that all the ot* *her arrows here are equivalences, as asserted. In particular, the map of towers P K* *(X)^ P L(X) ! P K,L(X) induces equivalences on all fibers, thus (inducting up the to* *wer) is itself a homotopy equivalence. 5.4. The Diagonal Theorem. Let r : K _ K ! K be the fold map. Since the diagonal on MapT (K, X) has a factorization MapT (K, X)_________________________//_MapT(K, X) x MapT (K, X), RR iiiiiii RRRRr*R iiiiiiiiii RRR iiiiiiiiiii RRR(( iiiiii MapT (K _ K, X) our Product Theorem has consequences for the diagonal map. Let : P K(X) ! P K(X) ^ P K(X) be the weak natural transformation * ~ P K(X) r--!P K_K(X) -!P K(X) ^ P K(X). Theorem 5.9. The weak natural transformation has the following properties. (1) There is a commutative diagram of weak natural transformations: eK (X) 1 MapT (K, X)+ ______________________//_P K(X) || || fflffl| eK (X)^eK (X) fflffl| 1 MapT (K, X)+ ^ MapT (K, X)+ __________//_P K(X) ^ P K(X). (2) Via the natural weak equivalences FkK(X) ' MapS (K(k), X^k)h k and Y Fk(P K(X) ^ P K(X)) ' MapS (K(i), X^i)h i^ MapS (K(j), X^j)h j, i+j=k 30 AHEARN AND KUHN the map induced by on kth fibers is the product, over i + j = k, of the composites of the weak natural transformations Tr kix j MapS (K(k), X^k)h k------! MapS (K(k), X^k)h( ix j) ß*-!Map (i) (j) ^k S(K ^ K , X )h( ix j) ~- Map (i) ^i (j) ^j S (K , X )h i^ MapS (K , X )h j, where ß : K(i)^ K(j)! K(k)is the projection. Property (1) follows immediately from property (1) of Theorem 5.5. To see that (2) follows from Theorem 5.5(2), we first observe that there is a factorization ________ß___________//(k) K(i)^ K(j)N 9K9 , NNN r(k)ssss NNNN ssss NNN'' ss (K _ K)(k) and thus a commutative diagram Map Sk(K(k), X^k)__________//_MapSix(jK(k), X^k) |r*| |ß*| fflffl| fflffl| MapSk((K _ K)(k), X^k)____//MapSix j(K(i)^ K(j), X^k). Thus Theorem 5.5(2) implies that the map on fibers can be identified with the product, over i+j = k, of the right vertical composites in the commutative diag* *rams of weak natural transformations Map S(K(k), X^k)h_k________~___________//MapSk(K(k), X^k) |Tr| || fflffl| |fflffl Map S(K(k), X^k)h( ix_j)_____~_________//_MapSix(jK(k), X^k) |ß*| |ß*| fflffl| |fflffl MapS(K(i)^ K(j),OX^k)h(Oix_j)__~_______//_MapSix(jK(i)^OK(j),OX^k) |o| |o| | | MapS (K(i), X^i)h i^ MapS(K(j), X^j)h~j^//_MapSi(K(i), X^i) ^ MapSj(K(j), X^j). 6.Compatibility results The following propositions and corollaries state that our various natural tra* *ns- formations are compatible in the expected ways. The three propositions are easi* *ly verified by directly checking their definitions. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 31 Proposition 6.1. For all J, K, and L, the diagram of natural transformations of towers 1^j ø K ^ ~P K^L(X)____//_K ^ ~P J^K^L(J ^_X)~_//K ^ ~P K^J^L(J ^ X) |ffl| |ffl| fflffl| j fflffl| P~L(X) ____________________________________//_~P(J^LX) commutes, where ø is the isomorphism induced by the twist map J ^ K ! K ^ J. Proposition 6.2. For all J, K, and L, the diagram of weak natural transforma- tions of towers j _________ P K_L(X)________//P J^(K_L)(J ^ X)_______P (J^K)_(J^L)(J ^ X) o~|| o|~| fflffl| j^j fflffl| P K(X) ^ P L(X)_________________________//P J^K(J ^ X) ^ P J^L(J ^ X) commutes. Corollary 6.3. For all J and K, the diagram of weak natural transformations of towers j P K(X) _____________________//_P J^K(J ^ X) || || fflffl| j^j fflffl| P K(X) ^ P K(X)__________//_P J^K(J ^ X) ^ P J^K(J ^ X) commutes. Example 6.4. A consequence of this corollary is that the truth of Corollary 1.7, for all n < 1, implies that Corollary 1.7 is true when n = 1. Proposition 6.5. For all J, K, and L, the diagram of weak natural transforma- tions of towers J ^ ~P J^(K_L)(X)_______________ffl______________//~P K_L(X) | ||^1 || fflffl| | J ^ J ^ ~P (J^K)_(J^L) o|~ | | | o|1^~ | fflffl| 1^ø^1 ffl^ffl fflffl| J ^ J ^ ~P J^K^ P J^L~__//J ^ ~P J^K^ J ^ ~P_J^L//_~P(KX) ^ ~P(LX) commutes, where is the diagonal and ø is the twist isomorphism. Corollary 6.6. For all J and K, the diagram of weak natural transformations of towers J ^ ~P J^K(X)__________________ffl________________//~P(KX) ||^ || fflffl| 1^ø^1 ffl^ffl fflffl| J ^ J ^ ~P J^K^ ~P J^K~__//J ^ ~P J^K^ J ^ ~P_J^K//_~P(KX) ^ ~P K(X) 32 AHEARN AND KUHN commutes, where is the diagonal and ø is the twist isomorphism. Example 6.7. With J = S1, a typical consequence of this would be the following. Let {E*,*r(K, X)} be the spectral sequence obtained from P K(X) by applying a cohomology theory with products. Then, because the reduced diagonal : S1 ! S1 ^ S1 is null, one deduces that Er(ffl) : E*,*+1r(K, X) ! E*,*r( K, X) is zero on the algebra decomposables in E*,*r(K, X). If the cohomology theory s* *at- isfies a Kunneth theorem (e.g. it is ordinary cohomology with field coefficien* *ts), the pinch map on S1 shows that E*,*r( K, X) is a Hopf algebra, and formal ma- nipulations then also imply that the image of Er(ffl) is contained in the primi* *tives. 7.Little cubes and an explicit S-duality map 7.1. Basic constructions with little cubes. Let I be the interval [-1, 1], and let C(n, 1) be the space of `little n-cubes', the space of embeddings In ! In w* *hich are products of n affine orientation preserving maps from I to itself. Then C(n* *, k) is defined to be the subspace of C(n, 1)k consisting of k-tuples of little n-cu* *bes whose images have disjoint interiors. Thus a point c 2 C(n, k) can be viewed an ` k O O embedding c : i=1In!In of a special form. Given a space Z, let F (Z, k) Zk denote the configuration space of k distin* *ct points in Z: F (Z, k) = {(z1, . .,.zk) | zi6= zj ifi 6= j}. It is well known and easy to prove that the map O C(n, k) ! F (In, k), sending a k-tuple of little n-cubes, (c1, . .,.ck), to their centers, (c1(0), .* * .,.ck(0)), is a k-equivariant homotopy equivalence. A point in C(n, k) provides a tubular neighborhood around the 0-dimensional O submanifold of In, consisting of the k center points. This suggests the follow* *ing ` k O O construction. Given a point in C(n, k), c : i=1In-!In, let `k c* : Sn -! Sn i=1 be the associated Thom-Pontryagin collapse map. Then define k` ff(n, k) : C(n, k)+ ! MapT (Sn, Sn) i=1 by ff(n, k)(c) = c*. Note that this is k-equivariant. The maps ff(n, k) are the starting points for two other families of maps. Let ffi(n, 1) : C(n, 1)+ ^ Sn ! Sn be the adjoint of ff(n, 1). Then notice th* *at the subspace C(n, k)+ ^ k(Sn) C(n, k)+ ^ Snk is sent to the basepoint by the map ffi(n, 1)^k : C(n, k)+ ^ Snk C(n, 1)k+^ Snk ! Snk. Thus ffi(n, 1)^k induces a k-equivariant map ffi(n, k) : C(n, k)+ ^ Sn(k)! Snk. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 33 A second family of k-equivariant maps fi(m, n, k) : Sm ^ C(m + n, k)+ ! Smk ^ C(n, k)+ is then defined by the following diagram: fi(m,n,k) Sm ^ C(m +"n,`k)+_____________//_Smk ^"C(n,`k)+ ||^i || fflffl| (ffi(m,1)^kOø)^1 fflffl| Smk ^ C(m, 1)k+^ C(n, 1)k+_______//_Smk ^ C(n, 1)k+, where ø : Smk ^ C(m, 1)k+' C(m, 1)k+^ Smk is the switch map, : Sm ,! Smk is the diagonal, and i : C(m + n, k) ,! C(m, 1)k x C(n, 1)k is the map which regar* *ds each little (m + n)-cube as the product of a little m-cube with a little n-cube. 7.2. The duality theorem and consequences. The following duality theorem will be proved in x7.3. Theorem 7.1. The map ffi(n, k) : C(n, k)+ ^ Sn(k)! Snk is an equivariant S- duality pairing. In other words, the stable adjoint ~ffi(n, k) : 1 C(n, k)+ ! MapS (Sn(k), Snk) is a homotopy equivalence of k-spectra. Given a space X, the map ffi(n, k) induces a natural map MapS (Snk, X^k) ! MapS (C(n, k)+ ^ Sn(k), X^k). A consequence of the duality theorem is that the adjoint of this, (7.1) C(n, k)+ ^ MapS (Snk, X^k) ! MapS (Sn(k), X^k), is a weak equivalence of k-spectra. Passing to homotopy orbits, we have con- structed the natural weak equivalence (1.2) of the introduction: n nk ^k FkS (X) ' (C(n, k)+ ^ MapS (S , X ))h k. Using [LMMS , Thm.I.7.9 and Prop.VI.5.3], we then deduce (1.3): n -n ^k FkS (X) ' C(n, k)+ ^ k ( X) . Assuming the duality theorem, we now deduce the corollaries of the introducti* *on from the corresponding theorems. Proof of Corollary 1.2.We specialize the Smashing Theorem to the case when K = Sn and L = Sm . We need to show that, under the equivalence (7.1), the descript* *ion of the map on fibers given in the theorem, corresponds to the description given* * in the corollary. 34 AHEARN AND KUHN By formal manipulation of adjunctions, we are asserting that there is a commu- tative diagram of k-spectra: ~ffi(m,k) C(m, k)+________~________//_MapS(Sm(k), Smk ) | | | j| | fflffl| i|| MapS (Sm(k)^ Snk, Smk ^ Snk) | | |* | p| fflffl| ~ffi(m+n,k) fflffl| C(m + n, k)+_____~_____//_MapS((Sm+n )(k), (Sm+n )^k). Here i is the inclusion induced by multiplying all little m-cubes by the iden* *tity n-cube. Again adjointing, we just need to check that there is a commutative diagram of k-spaces: C(m, k)+ ^ (Sm+n )(k)__i^1____//C(m + n, k)+ ^ (Sm+n )(k). | | | | 1^p|| |ffi(m+n,k)| | | | | fflffl| ffi(m,k)^1 fflffl| C(m, k)+ ^ Sm(k)^ Snk___________//Smk ^ Snk = (Sm+n )^k This diagram, in turn, is a quotient of smash products of the diagram in the ca* *se when k = 1. This is the diagram ffi(m,1)^1 C(m, 1)+ ^ Sm+n_______//_Sm+n QQ OO QQQQiQ | QQQ |ffi(m+n,1) QQQ(( | C(m + n)+ , which is easily verified to be commutative. Proof of Corollary 1.7.The corollary follows from the Diagonal Theorem, once we verify that, for all i + j = k, there is a commutative diagram of ix j-spectr* *a: ~ffi(n,k) C(n, k)+__________~___________//MapS(Sn(k), Snk) | | | |ß* | fflffl| |i| MapS (Sn(i)^ Sn(j), Snk) | OO | |~ | | fflffl| ~ffi(n,i)^~ffi(n,i) | (C(n, i) x C(n, j))+_~_____//_MapS(Sn(i), Sni) ^ MapS (Sn(j), Snj) Here i is the inclusion which sends a k-tuple of little cubes to the first i cu* *bes and the last j cubes. This diagram of spectra commutes because there is a commutative diagram of ix j-spaces: STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 35 C(n, k)+ ^ Sn(i)^ Sm(j)_i^1____//C(n, i)+ ^ C(n, j)+ ^ Sn(i)^ Sn(j) | | |1^ø^1 | | | fflffl| 1^ß| C(n, i)+ ^ Sn(i)^ C(n, j)+ ^ Sn(j) | | | | |ffi(n,i)^ffi(n,j) fflffl| ffi(n,k) fflffl| C(n, k)+ ^ Sn(k)___________________//Snk = Sni^ Snj. Here ø is the twist map. Proof of Corollary 1.9.The corollary follows from the Evaluation Theorem, once we verify that there is a commutative diagram of k-spectra: 1^~ffi(m+n,k) Sm ^ C(m + n, k)+____~_____//Sm ^ MapS ((Sm+n )(k), (Sm+n )^k) | | | |1^d* | fflffl| fi(m,n,k)|| Sm ^ MapS (Sm ^ Sn(k), (Sm+n )^k) | | | | |ffl fflffl| ~ fflffl| Smk ^ C(n, j))+_,_____________//_MapS(Sn(k), Smk ^ Snk) where the unlabelled horizontal arrow is adjoint to 1 ^ ffi(n, k) : Smk ^ C(n, k)+ ^ Sn(k)! Smk ^ Snk. This diagram of spectra commutes because there is a diagram of k-spaces: ffi(m+n,k) C(m + n, k)+O^O(Sm+n )(k)________//(Sm+n )^kO=OSmk ^ Snk | 1^d|| | | || C(m + n, k)+O^OSm ^ Sn(k) 1^ffi(n,k)|| ø^1|| || | fi(m,n,k)^1 | Sm ^ C(m + n, k)+ ^ Sn(k)________//_Smk ^ C(n, k)+ ^ Sn(k), easily checked to be commutative. 7.3. Proof of the duality theorem. Our strategy in proving Theorem 7.1 is to show our map is equivariantly homotopic to a duality map constructed in a stand* *ard way. We begin with a general equivariant duality construction. Let V be a real rep- resentation of a finite group G, and let SV denote the one point compactificati* *on V [ {1}, regarded as a based G-space with basepoint 1. Let ~ : V+ ^ SV ! SV be defined by ( ~(x, y) = x - y ifx, y 2 V 1 otherwise. 36 AHEARN AND KUHN This is a well-defined continuous G-map. Let K SV be a based G-subspace such that (SV , K) is an equivariant NDR pair. (Equivalently, the inclusion of K into SV is an equivariant cofibration.)* * The map ~ induces a map of pairs: ~ : (SV - K)+ ^ (SV , K) ! (SV , SV - 0). The following is presumably well known. Proposition 7.2. ~ is an S-duality map, in the sense that (SV - K)+ ^ (SV , K) ~-!(SV , SV - 0) -~ (SV , 1) induces a duality map ~~: (SV - K)+ ^ SV =K ! SV . Sketch Proof.In the nonequivariant case, this can be read off of Spanier's orig* *inal paper [Sp]. For the equivariant case, apply [LMMS , Construction III.4.5] to * *the following situation: let N SV be a G-neighborhood of K such that K N is an equivariant homotopy equivalence, then let X = SV - N and A = ;. Via the equivalences C(V, V - X) ' SV =K and C(X, ;) ' (SV - K)+ , the map this construction yields corresponds to ~~. Example 7.3. This construction gives us k - duality maps ~(n, k) : F (Rn, k)+ ^ (Snk, k(Sn)) ! (Snk, Snk - 0). Now consider the following situation. Suppose given ` : Rn+^ Sn ! Sn satisfying the condition (7.2) `(x, y) = 0 only ifx = y. Then `^k : Rnk+^ Snk ! Snk restricts to define a k-equivariant map `(k) : F (Rn, k)+ ^ (Snk, k(Sn)) ! (Snk, Snk - 0). Example 7.4. If ~(n) : Rn+^ Sn ! Sn is defined by ( n ~(n)(x, y) = x - y ifx, y 2 R 1 otherwise, the resulting map ~(n, k) : F (Rn, k)+ ^ (Snk, k(Sn)) ! (Snk, Snk - 0) is precisely the duality map of Example 7.3. A little variation on this last construction goes as follows. Suppose given ` : C(n, 1)+ ^ Sn ! Sn satisfying the condition (7.3) `(c, y) = 0 only ifc(0) = y. (Recall that c(0) is the center of the little cube c.) Then `^k : C(n, 1)k+^Snk* * ! Snk restricts to define a k-equivariant map `(k) : C(n, k)+ ^ (Snk, k(Sn)) ! (Snk, Snk - 0). STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 37 O ~ Example 7.5. Choose a homeomorphism h :In-! Rn. Then define d(n) : C(n, 1)+ ^ Sn ! Sn to be the composite O h^1 ~(n) C(n, 1)+ ^ Sn !In+ ^Sn --! Rn+^ Sn ---! Sn, where the first map sends a little cube to its center, and ~(n) is as in Exampl* *e 7.4. The resulting family of maps, d(n, k) : C(n, k)+ ^ (Snk, k(Sn)) ! (Snk, Snk - 0), will be duality maps, as the maps ~(n, k) were. Example 7.6. Let ffi(n) : C(n, 1)+ ^ Sn ! Sn be defined by ffi(n)(c, y) = c*(y). Then (7.3) holds, and the resulting maps ffi(n, k) : C(n, k)+ ^ (Snk, k(Sn)) ! (Snk, 1) ,! (Snk, Snk - 0) are the maps of x7.1. Let id 2 C(n, 1) denote the identity cube. Lemma 7.7. Suppose ` : C(n, 1)+ ^ Sn ! Sn satisfies (7.3) and also (7.4) `(id, y) = y for all y in Sn. Then `(k) is equivariantly homotopic to the map d(n, k) of Example 7.5. Momentarily assuming this, Theorem 7.1 follows: since the map ffi(n) of the Example 7.6 satisfies the hypotheses of the lemma, we conclude that ffi(n, k) is homotopic to the known duality map d(n, k). Proof of Lemma 7.7.A map ` satisfying (7.3) can be regarded as a map of pairs (C(n, 1)+ ^ Sn, C(n, 1)+ ^ Sn - {(c, y) | c(0) = y}) `-!(Sn, Sn - 0). Now we observe that the inclusion Sn ,! C(n, 1)+ ^ Sn sending y to (id, y) indu* *ces a homotopy equivalence of pairs i : (Sn, Sn - 0) ~-!(C(n, 1)+ ^ Sn, C(n, 1)+ ^ Sn - {(c, y) | c(0) = y}). Thus maps ` satisfying (7.3) are classified up to homotopy by the homotopy clas* *s of ` Oi : (Sn, Sn -0) ! (Sn, Sn -0). But condition (7.4) precisely says that ` Oi * *is the identity map, as is d(n) O i. Thus ` is homotopic to d(n) through maps satisfyi* *ng (7.3), and so `(k) is homotopic to d(n, k) for all k. 8.The operad action theorem In this section we use some of the ideas from the previous section to state a* *nd prove a more precise version of Theorem 1.10. Firstly, the structure map `(r) : C(n, r) x r ( nX)r ! nX, ` is easy to define. Since ( nX)r = MapT ( Sn, X), `(r) is the map induced by r ` ff(n, r) : C(n, r)+ ! MapT (Sn, Sn). r 38 AHEARN AND KUHN Next we observe that contravariant functor from spaces to towers of spectra sending K to P K(X) is continuous, and that eK (X) is a natural transformation * *of continuous functors. Thus one gets maps MapT (K, L) ! Map S(P L(X), P K(X)), natural in all variables, and compatible with MapT (K, L) ! MapT (MapT (L, X), MapT (K, X)). Adjointing the r-equivariant composite ` W n n C(n, r)+ -ff(n,r)---!MapT(Sn, Sn) ! Map S(P rS (X), P S (X)) r yields a natural maps of towers W n n `(r) : C(n, r)+ ^ r P rS (X) ! P S (X). This is the map of Theorem 1.10, and property (1) listed there clearly holds. It remains to identify the map on kthfibers in terms of the little n-cubes op* *erad structure. One approach to the operad structure is as follows. Let C(n, r, k) be the spa* *ce of all embeddings ak O ar O c : In! In i=1 j=1 such that each nontrivial component is a little n-cube. This is a r x k-space. Then the operad structure is given by the k-equivariant maps Or,k: C(n, r) x r C(n, r, k) ! C(n, k) sending (c, d) to the composition c O d. To see that this agrees with the definition given in [M ], note that C(n, r, * *k) decomposes as the product, over all maps ~ : k ! r, of the spaces C(n, k1) x . .x.C(n, kr), where kj = |~-1(j)|, and the corresponding component of Or,kis the usual struct* *ure map. W n W The kthfiber of P rS (X) is naturally equivalent to MapS (( rSn)(k), X^k)h * *k, and, by construction, the map `(r, k) induced by `(r) on kth fibers is induced * *in the apparent way by ff(n, r). Generalizing definitions in x7, let ` ffi(n, r, 1) : C(n, r, 1)+ ^ ( Sn) ! Sn r W be the map sending (c, t) to c*(t), where c* : r Sn ! Sn is the Thom-Pontryagin collapse map associated to c. As before, ffi(n, r, 1)^k induces maps ` ffi(n, r, k) : C(n, r, k)+ ^ ( Sn)(k)! Snk. r Using Theorem 7.1, one can deduce that the stable adjoint of ffi(n, r, k), ~ffi(n, r, k) : 1 C(n, r, k)+ ! MapS ((` Sn)(k), Snk) r STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 39 is an equivalence. There results a natural weak equivalence of r-spectra: ` C(n, r, k)+ ^ k ( -n X)^k ' MapS (( Sn)(k), X^k)h k. r We note that it is easyWto see that this equivalence is compatible with the d* *e- composition of MapS (( rSn)(k), X^k)h k arising in our product theorems. Property (2) of Theorem 1.10 can now be more precisely stated. Proposition 8.1. There is a commutative diagram of weak natural transforma- tions: C(n, r)+ ^ r C(n, r, k)+ ^ k ( -nX)^k~//_C(n, r)+ ^ r MapS((W rSn)(k), X^k)h k |Or,k^1| `(r,k)(X)|| fflffl| ~ fflffl| C(n, k)+ ^ k ( -nX)^k_______________//_MapS(Sn(k), X^k)h k. Proof.It suffices to show that there is a commutative diagram of k-spaces 1x~ffi(n,r,k) W C(n, r) x r C(n, r,_k)_______//C(n, r) x r MapT (( rSn)(k), Snk) |Or,k| |ff| fflffl| ~ffi(n,k) fflffl| C(n, k)_______________________//_MapT(Sn(k), Snk) where ff is induced by ff(n, r). That this diagram commutes is easily checked: * *given c 2 C(n, r) and d = (d1, . .,.dk) 2 C(n, r, k), we have that (ff O (1 x ~ffi(n, r, k)))(c, d) = (d*1O c*) ^ . .^.(d*kO c*), while (~ffi(n, k) O (Or,k))(c, d) = (c O d1)* ^ . .^.(c O dk)*. These agree because the Thom-Pontryagin collapse is a contravariant functor: (c O d)* = d* O c*. Appendix A. A proof of Arone's theorem when K is a sphere In this appendix, we give a proof of Theorem 3.6 in the case when K is a sphe* *re. The first reductions follow along the lines of Arone's proof in [Ar], but we use Theorem 7.1 to simplify the proof of the last key step: proving that a certain * *`cross effect' map is an equivalence. It seems likely that some variant of our proof c* *an be used to prove the theorem in general. As does Arone, we use ideas from Goodwill* *ie calculus at a couple of points. The theorem we are trying to prove says that n 1 n Sn eS1(X) : MapT (S , X)+ ! P1 (X) is a weak homotopy equivalence if the connectivity of X is greater than n. To explain why this is equivalent to Theorem 3.6, and to effectnour first red* *uction, let F (X) and G(X) respectively denote the domain and range of eS1(X). [G2 , Example 4.5] says that F (X) is n-analytic: this means that F behaves well (in a precise sense defined in [G2 ]) on n-connected spaces. Consideration of the fib* *ers of the Arone tower, identified in Proposition 3.9, shows that the projection maps n Sn qk : G(X) = P1S (X) ! Pk (X) 40 AHEARN AND KUHN are (1 + connX - n)(1 + k) - 1 connected, and then that G(X) is n-analytic.n Goodwillie's argument proving [G2 , Prop.5.1] then shows that eS1(X) will be a weak equivalence for all n-connected X if it is an equivalence for all X of the* * form nY , with Y connected. At this point, we use the classical model Cn(Y ) for MapT (Sn, nY ) built fr* *om the spaces C(n, k). For a space Y with basepoint *, let a1 Cn(Y ) = ( C(n, k) x k Y k)=(~ ), k=1 where (c1, . .,.ck, y1, . .,.yk-1, *) ~ (c1, . .,.ck-1, y1, . .,.yk-1) generate* *s the equiv- alence relation. Cn(Y ) is filtered, with ak FkCn(Y ) = ( C(n, j) x j Y j)=(~ ), j=1 and there are cofibration sequences Fk-1Cn(Y ) ! FkCn(Y ) ! C(n, k)+ ^ k Y ^k. A natural map ffn(Y ) : Cn(Y ) ! MapT (Sn, nY ) is defined to be the map induced by the maps ` ` O C(n, k) x Y k-ff(n,k)xj-----!MapT(Sn, Sn) x MapT ( Sn, nY ) -!MapT (Sn, n* *Y ). k k ` O O Explicitly, if we regardWc 2 C(n, k) as an embedding c : kIn- !In, and y 2 Y k as a based map y : k S0 ! Y , then ffn(Y )([c, y]) = ( ny) O c*. The classical theorem, [M , Thm.2.7], then states that ffn(Y ) is a weak homoto* *py equivalence if Y is connected.n It follows that, to show eS1( nY ) is an equivalence, it suffices to show that n n `(n, k, Y ) : 1 FkCn(Y )+ ! PkS ( Y ) is an equivalence, where `(n, k, Y ) is the composite Sn( nY ) n 1 FkCn(Y )+ ,! 1 Cn(Y )+ -ffn(Y-)--! 1 MapT (Sn, nY )+ -ek------!PkS ( nY ). This we proceed to show by induction on k, using ideas from [G3 ]. If G : T !* * S is a functor, we let pkG : T ! S be its universal k-excisive quotient, and ØkG : T k! S its kth cross effect. It is quite easy to see that pk preserves fibration sequences of functors, an* *d, since our functors take values in spectra, also cofibration sequences. Also, the fun* *ctor C(n, k)+ ^ kY ^kis an example of a homogeneous functor of degree k, i.e. a func* *tor G with G ' pkG and pk-1G ' *. n Such considerations show thatnbothn 1 FkCn and PkS are k-excisive, and both i : Fk-1Cn ,! FkCn and p : PkS ! PkS-1induce equivalences after applying pk-1. STRUCTURE IN RESOLUTIONS OF MAPPING SPACES 41 Then the inductive hypothesis, combined with the commutativity of the diagram `(n,k,Y ) n 1 FkCn(YO)+____________//PkSO( nY ) |i| |p| | `(n,k-1,Y ) fflffl|n 1 Fk-1Cn(Y )+___________//PkS-1( nY ) shows that pk-1`(n, k, Y ) is an equivalence for all Y . [G3 , Lemma 3.2] implies that a natural map ` between k-excisive functors will be an equivalence if both pk-1` and Øk` are equivalences. Thus, our inductive proof that `(n, k, Y ) is an equivalence will be complete if we establish that n n n Øk`(n, k)(Y1, . .,.Yk) : Øk 1 FkCn(Y1, . .,.Yk) ! ØkPkS ( Y1, . .,. Yk) is an equivalence. Cross effects are defined as iterated fibers of certain cubical diagrams. But* * if a functor G takes values in spectra (where finite coproducts are equivalent to fi* *nite products), ØkG(Y1, . .,.Yk) is naturally equivalent to the iterated cofiber of * *the k- dimensional cube with entries G(Z1_. ._.Zk) with Zj 2 {*, Yj}, and the canonical map G(Y1 _ . ._.Yk) ! ØkG(Y1, . .,.Yk) is a retraction. It is immediate from the definitions that, when G = 1 FkCn, this retraction * *is the map 1 FkCn(Y1 _ . ._.Yk)+ ! 1 C(n, k)+ ^ (Y1 ^ . .Y.k) induced by the evident map C(n, k)+ x k (Y1 _ . ._.Yk)k ! C(n, k)+ ^ (Y1 ^ . .^.Yk). Now note that the functor æ : SEk ! S k, which sends a functor X to the k-spectrum Xk=Xk-1(k), induces a natural map æ : Map EkS(K^ , X^ ) ! Map Sk(K(k), X(k)) such that Map Tn(K(k), X^k) i-!MapEkS(K^ , X^ ) æ-!MapS k(K(k), X(k)) is induced by ß : X^k ! X(k). It follows that, when G = PkSnO n, the retraction is equivalent to the composite Map EkS(S^, ( nY1 _ . ._. nYk)^)æ-!MapSk(Sn(k), ( nY1 _ . ._. nYk)(k)) ! MapS (Sn(k), nY1 ^ . . .nYk). If Y is a space, let ~ffi(n, k, Y ) : C(n, k)+ ^ Y ! MapS (Sn(k), Snk ^ Y ) b* *e the map adjoint to ffi(n, k) ^ 1Y . By Theorem 7.1, ~ffi(n, k, Y ) will be a weak equiv* *alence. Lemma A.1. The diagram 1 C(n, k)+ x k (Y1_ . ._.Yk)k`(n,k)(Y1_..._Yk)//_MapEkS(S^, ( nY1_ . ._. nYk* *)^) | | | | fflffl| ~ffi(n,k,Y1^...^Yk) fflffl| 1 C(n, k)+ ^ (Y1^ . .^.Yk)_____________//MapS(Sn(k), nY1^ . .^. nYk) commutes. 42 AHEARN AND KUHN Assuming this, our proof of Theorem 3.6 is done, as we can identify the map Øk`(n, k)(Y1, . .,.Yk) with the weak equivalence ~ffi(n, k, Y1 ^ . .^.Yk). The functor T ! T sending Y to Y (k)is continuous. Thus it induces natural maps ,(k): MapT (K, Y ) ! MapT (K(k), Y (k)). Recalling the definition of the m* *aps eKk(X), one sees that the commutativity of the diagram in the lemma will follow from the commutativity of the following diagram of spaces: C(n, k)+ x (Y1_ . ._.Yk)kff(n,k)xj//_MapT(Sn, WkSn) x MapT(W kSn, nY1_ . ._. * *nYk) | | O| | | | fflffl| | MapT(Sn, nY1_ . ._. nYk) | | |(k) | ,| | fflffl| | MapT(Sn(k), ( nY1_ . ._. nYk)(k)) | | | | | fflffl| ~ffi(n,k,Y1^...^Yk) fflffl| C(n, k)+ ^ (Y1^ . .^.Yk)_____________//MapT(Sn(k), nY1^ . . .nYk). The commutativity of this diagram is verified easily. That the two maps agree on an element (c1, . .,.ck, y1, . .,.yk), with ci 2 C(n, 1) and yi 2 Yi (viewed* * as a map yi: S0 ! Yi), amounts to the observation that the diagram _c^k_//W n ^k_(_n(y1_..._yk))^k//_n n ^k Snk ( kS ) ( Y1 _ . ._. Yk) |c*1^...^c*k| || fflffl| ny1^...^ nyk fflffl| Snk_________________________________// nY1 ^ . .^. nYk commutes. Remark A.2. The argument here shows that the natural transformations n n 1 n n Sn n eSk( Y ) : ( Y )+ ! Pk ( Y ) and n n n pSk( nY ) : PkS ( nY ) ! PkS-1( nY ) admit compatible natural weak right inverses n n 1 n n sk : PkS ( Y ) ! ( Y )+ and n n tk : PkS-1( nY ) ! PkS ( nY ). This is a form of the Snaith splitting theorem, and the splitting obtained this* * way is equivalent to other classical constructions [K2 , Appendix B]. References [Ad]J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's * *lecture, Algebraic topology (Aarhus, 1982), Springer L. N. 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Spanier, Function spaces and duality, Ann. Math. 70(1959), 338-378. [WW] M. Weiss and B. Williams, Automorphisms of manifolds and algebraic K-theor* *y, part II, J. P. A. A. 62(1989), 47-107. [Wh]G. W. Whitehead, Elements of homotopy theory, Springer Graduate Text in Mat* *h. 61, 1978. Department of Mathematics, De Pauw University, Greencastle, IN 46135 E-mail address: sahearn@depauw.edu Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: njk4x@virginia.edu