On vanishing Tate cohomology and decompositions in Goodwillie calculus Kristine Bauer Randy McCarthy *y March 24, 2003 Abstract Our main result is that if F is a functor from a pointed category C to spectra, the Goodwillie tower of F evaluated at X splits rationally when X is a co-H-object of C. We show that the layers of F (X) in this case are easy to identify. The splitting of the Goodwillie tower gives a decomposition of F (X) into a product of its layers. We use this to recover the rational decompositions of Hochschild and higher Hochschild homology [P00 ], [L98 ], [GS87 ]. Finally, we extend the main theorem to include dual calculus to recover the Poincar'e-Birkhoff-Witt theorem, and improve the theorem in the special case in which the comultiplication map is cocommutative. 1 Introduction Let F be a functor from any pointed category C to any abelian category. As an application of [G90 ], [G92 ] and [G02 ], Johnson-McCarthy provide a ______________________________ *The second author was partially supported by NSF grant DMS0071482 1 Goodwillie calculus theory for such functors ([JM4 ]). There is a tower 6..6 mmmmmm. mm mmmmmm |qn+1| mmmm pn fflffl| F (X) ___________//_RRPnF (X) RRRRpn-1RR | RRR |qn RR((R fflffl| Pn-1F (X) | | fflffl| .. . of universal degree n approximations PnF (X) to F (X). The layers DnF (X) of F (X) are the fibers of the maps qn. This model of calculus can be extended to functors F from C to spectra, as in [McC02 ] (by dualizing). In this paper, we provide a criterion on X for the Goodwille tower of F to split rationally when evaluated at X. In particular, we show that if X is an object with a öc multiplication" map r from X to the coproduct X _ X - in other words, if X is a co-H-object in C - then rationally the first map of the fiber sequence qn DnF (X) _____//PnF (X)____//_Pn-1F (X) has a splitting map. This results in the main theorem of the paper: Theorem 1.1. If F is a homotopy functor and X is a co-H-object of C then rationally Yn PnF (X) ' DiF (X) i=1 Q and consequently P1 F (X) ' n 1DnF (X). The proof of the main theorem is constructive. We construct maps PnF(r) DnF (X) _____//PnF (X)____//_PnF (_nX)___//_DnF (X) such that the composite is multiplication by n!. To obtain the splitting, we simply invert n! (thus the result is rational). The bulk of the proof involves 2 showing that this composite map is induced by the norm map (x3), which in turn induces the Tate map and shows that the composite is multiplication by n!. Since X is a co-H-object, it is equipped with öc vering maps" r defined by __rr_//_X_+___// X r X where + is the fold map. An important consequence of the proof of the main theorem is that the maps r induce multiplication by rn on DnF (X), and hence can be used to identify the layers of F (X). We use this idea to show that the known rational decompositions of (higher) Hochschild homology ([GS87 ], [L98 ], [P00 ], [B ]) are actually splittings of the Goodwillie tower* * of the forgetful functor from augmented commutative k-algebras to k-modules. We then compute the layers of the tower associated to higher Hochschild homology, and show that they are suspensions of the layers of the tower associated to Hochschild homology. We also prove a version of the main theorem for dual calculus ([McC02 ]). In this case, we examine the H-objects of C. Using the dual version of our main theorem, we are able to show that rationally, the Poincar'e-Birkhoff- Witt theorem is a decomposition of the universal enveloping algebra of a free Lie algebra into the dual layers of a dual Goodwillie tower associated to it. Finally, we show that sometimes the Goodwillie tower splits more often than är tionally". Recall that the Tate map provides a map from the homo- topy orbits to the homotopy fixed points of a spectrum. We are particularly interested in the Tate map applied to the spectrum D(n)1crnF (X), the multi- additivitation of the n-th cross effects of F (see section 2 for the definition* * and close relationship to DnF ). Define the n-th Tate cohomology of F evaluated at X to be the cofiber of this map applied to crnF (X). In the final section of the paper, we show that if the comultiplication map is cocommutative, then the Goodwillie tower actually splits whenever the Tate cohomology vanishes. Kuhn has recently observed that work of Greenlees, Hovey and Sadofsky shows that the Tate cohomology of a K(n)*-local spectrum vanishes - hence the Goodwillie calculus towers of functors from cocommutative co-H-objects to spectra split upon K(n)*-localization. This paper is organized as follows. In section 2 we define and examine the cross effects of a functor. It is important to note that in this section we do not construct the universal degree n approximations to F . However, we do 3 provide all of the results and constructions that we will need in the context of this paper. A more detailed account can be had from [JM4 ] and [M ] (for the case where the target category is spectra) or [McC02 ]. In section 3 we describe the norm map and its relationship to the Tate map, which will be essential in constructing the splitting map to DnF (X) ! PnF (X). Here we only provide statements of the results we will need regarding these maps; a more detailed account of the Tate map is given in the appendix of the preprint version [BMc ]. In section 4 we make explicit our definition of co-H-objects. We also check that the properties we require of F are preserved by crn, Dn and Pn. Section 5 is the statement and proof of the main theorem. We conclude by giving two extensions of the main theorem; first the extension to dual calculus (section 6) and finally to the case of vanishing Tate cohomology (section 7). The authors would like to thank Nick Kuhn for pointing out an error in an earlier draft of this paper. 2 Preliminaries in Goodwillie Calculus Let S be any category of rational spectra, that is, modules over the Eilenberg- MacLane spectrum HQ. Let F be a functor from a pointed category C to S. The building blocks used to construct the universal degree n approximation to F are called the cross effects. In this paper, we will usually phrase all important information about the Goodwillie tower of F in terms of cross effects. For this reason, it is important for us to spend some time introducing them. It is easiest to think of the cross effects as the total fibers of certain cubical diagrams. Let n = {1, . .,.n} be a finite set of n elements. Let P (n) be the category whose objects are the subsets of n and whose morphisms are ordered inclusions. Note that the subset ; is an initial object of P (n) while n is the terminal object. One can visualize this category as a cubical diagram with an object of P (n) at each corner and morphisms as edges. The objects ; and n occupy opposite corners of the cube. An n-cube in a pointed category C (i.e. a category whose initial object is also its terminal object) with coproducts is a contravariant functor from P (n) to C. To construct the cross effects, we will use a particular n-cube. Let _ denote the coproduct in C. Let g : n ! obj(C) be a function which 4 generates a list of objects X1, . .,.Xn of C. Define an n-cube Øg to be the contravariant functor which is given on a subset S of n by Øg(S) = _ g(c) c=2S " and which takes an inclusion SØ____//_S0to the projection _ g(d) ! _ g(c). d=2S c=2S0 By convention, Øg(n) = * (the base point of C). For n = 2, Øg is the square diagram: X1 _ X2 _____//X1 | | | | fflffl| fflffl| X2 ________//* Let F be a covariant functor from C to S. Let P0(n) be the subcategory of P (n) consisting of non-empty subsets and let Xi = g(i). Definition 2.1. The n-th cross effect, crnF (X1, . .,.Xn), is the fiber of the map holimS2P(n)F (Øg(S)) ! holimS2P0(n)F (Øg(S)). Note that holimS2P(n)F (Øg(S)) = F (_ni=1Xi). The cross effect crnF (X1, . .,.Xn) is an n multi-variable functor from Cxn to S. Let crnF (X) = crnF (X, . .,.X). Notice that there is a convenient map æ : crnF (X) ! F (_nX) arising from the definition of the cross effect. Definition 2.2. A functor F : C ! S is a degree n functor if crkF ' * for every k > n. If F is degree n, then the Goodwillie tower of F is truncated - i.e. DnF is the largest non-trivial layer of F and PkF is equivalent to F for all k n. This is sometimes taken as the definition of degree n. Furthermore, if F is degree n, then crnF is linear in each variable. That is, e.g. crnF (X _ Y, X2, . .,.Xn) ' crnF (X, X2, . .,.Xn) x crnF (Y, X2, . .,.Xn). (1) We enumerate here some properties of the cross effects and Goodwillie towers of degree n functors which we will frequently use. 5 Lemma 2.3. If F is a degree n functor, then there exists a natural equiva- lence m Y crnF ( _ Xi) ' crnF (Xff(1), . .,.Xff(n)) (2) i=1 ff2Hom (n,m) Proof.This is a consequence of the fact that for degree n functors, crnF (X) __ is linear in each variable. |__| Remark 2.4. Let uk : _mi=1Xi ! Xk (1 k m) be the map which is the identity on the Xk component of _mi=1Xiand which sends Xj to the basepoint for j 6= k. The weak equivalenceQof Lemma 2.3 can be realized by the map ! : crnF (_mi=1Xi) ! ff2Hom (n,m)crnF (Xff(1), . .,.Xff(n)) defined by Y ! = crnF (uff(1), . .,.uff(n)) O ff2Hom (n,m) Q where is the diagonal map crnF (_mi=1Xi) ! Hom(n,m)crnF (_mi=1Xi). Let ck : Xk ! _mi=1Xi be the dual map to uk which includes the k-th summand. Keeping in mind that coproducts and products in S are weakly equivalent, we construct a homotopy inverse to ! given by the composition of ` ` crnF(cf(1),...,cf(n))` crnF (X) f2Hom(n,m)___________//_ crnF (_ X) f2Hom (n,m) f2Hom (n,m) m with the fold map. Lemma 2.5. If F is a degree n functor, then there exists a natural equiva- lence Y crncrnF (X1, . .,.Xn) ' crnF (Xff(1), . .,.Xff(n)). (3) ff2 n Proof.Let g(i) = Xi and let cS denote the complement in n of a subset S of n. Using the definition of the cross effect and lemma 2.3, we compute that crncrnF (X1, . .,.Xn) = fiber{holim crnF ( _ Xc) ! holim crnF ( _ Xc)} S2P(n) c2cS S2P0(n) c2cS 6 is equivalent to the fiber of Y Y holim crnF (Xff(1), . .,.Xff(n)) ! holim crnF (Xff(1), . .,.Xff(* *n)). S2P(n) ff2 S2P0(n) ff2 Hom(n,cS) Hom(n,cS) Q The first homotopy limit is just ff2Hom (n,n)crnF (Xff(1), . .,.Xff(n)) (since thisQlimit diagram has an initial object) and the second homotopy limit is ff2Hom 0(n,n)crnF (Xff(1), . .,.Xff(n)) where Hom 0(n, n) are the non-biject* *ive set maps from n to itself. So this is equivalent to the fiber of Y Y crnF (Xff(1), . .,.Xff(n)) ! crnF (Xff(1), . .,.Xff(n)) ff2Hom (n,n) ff2Hom 0(n,n) which is Y crnF (Xff(1), . .,.Xff(n)) ff2Bij(n,n) __ where Bij(n, n) := n is the group of bijective maps. |__| Remark 2.6. Using remark 2.4,Qone can realize the weak equivalence of 2.5 by a map !0: crncrnF (X) ! Bij(n,n)crnF (X) defined by Y !0= crnF (uff(1), . .,.uff(n)O O æ ff2Bij(n,n) where æ : crncrnF (X) ! crnF (_nX) is the natural map. This is accom- plished by applying the map ! to crnF (_c2cSXc) for each S 2 P (n) (or P0(n)). Proposition 2.7 (Lemma 3.9 [JM4 ]). If F is a degree n functor from C to S then DnF ' (crnF )h n. Proposition 2.8 (Remark 2.8 [JM4 ]). If F is a degree n functor from C to S then F ' PnF . By using fibers instead of cofibers in the definition of the cross effect, we obtain another cross effect, called the co-cross effect, ecrn. Let P1(n) denote the full subcategory of P (n) consisting of proper subsets of n. Let eØ be the covariant functor from P (n) to C with eØ(S) = _ g(c) and which takes c2S inclusions in P (n) to inclusions in C. 7 Definition 2.9. We define the co-cross effect ecrnF (X1, . .,.Xn), to be the cofiber of the map holimS2P1(n)F (eØg(S)) ! holimS2P(n)F (eØg(S)). In the category of spectra, since cofibration and fibration sequences are equivalent, we have that crnF ' ecrnF . There is a convenient map F (_nX) ! ecrnF (X) (dual to the map æ) which we will need. Lemma 2.10. If F is a degree n functor, then there exist natural equivalences a cernF (_ X) ' ecrnF (X) (4) n Hom (n,n) a ecrnecrnF (X)' ecrnF (X). (5) n Proof.This is the analogue of lemmas 2.3 and 2.5, and the proof is straight- __ forward. |__| 3 The Tate Map If a finite group G acts on a k-module M, there is a natural map t : MG ! MG induced by the norm map. The norm map is constructed using the diagonal , the action of g for each g 2 G, and the addition map + as follows: ______t0__________________________________________* *_________________________________________________________________@ __________________________________--________________________* *_________________________________________________________________@ ____//_ ____//_ +___//_ M |G|M g2Gg |G|M M We can extend this to a map t : MG ! MG by making two observations. First, t0extends in the following diagram (where p is the projection map onto the orbits) __t0_//_ M zM== p|| z z fflffl|z MG 8 since for each g 2 G, t0(m) = t0(gm).P Second, note that the image of t0 actually lands in MG since t0(m) = g2G gm so that for any h 2 G, X X X ht0(m) = h gm = hgm = gm = t0(m) g2G g2G g2G since G is finite. Thus we have a map t which factors t0 as: __t0__// M MOO p|| |i| fflffl|t ?Ø| MG ____//_MG where i is the inclusion of the fixed points. The map t is an equivalence whenever the order of G is invertible. In fact, the inverse is given by p O i and we have p O i O t[m] = |G|[m]. We wish to extend this map to the category S of HQ-modules. Suppose that E is an HQ-module with an action of a finite group G. We can extend the norm map T 0. Again, we do this by using the diagonal map and the fold map. This time, we must use the weak equivalence between the product and the coproduct. The following diagram defines T 0: Q Q gx(-) Q ` _____// E _g2G______//_E o'o_?_` E _+___// E g2G g2G g2G E Motivated by this diagram, we make the following definition Definition 3.1. Let I be an indexing set and {Xi}i2I be a collection of objects of S. Let fi,j: Xi ! Xj be maps in S for each i, j 2 I and gi,j: Xio'o__Xj be weak equivalences in S for each i, j 2 I. A weak map in S from Xim ! Xin is a collection of objects Xim, Xim+1, . .,.Xim+k = Xin and arrows fi,jand gi,jsuch that for any adjacent pair Xim+j, Xim+j+1of ob- jects in the collection, there is either a map fim+j,im+j+1or a weak equivalence gim+j,im+j+1between them. A typical weak map might look like a zig-zag fi,j gj,k Xi _____//Xjo'o_Xk. 9 g-1j,kOfi,j This weak map is denoted Xi ` ` ` ` ` `//Xj, where the dashed line denotes that it's a weak map. Note that each weak map has associated to it a map in the homotopy category. We say that a diagram of weak maps commutes if the corresponding diagram of maps in the homotopy category commutes. There is a weak map T which extends the weak map T 0to homotopy orbits and homotopy fixed points just as t extended t0 in the case of modules. This weak map is called the Tate (weak) map. A construction of t can be found in e. g. [McC ]. We will need the following properties of the Tate map. The proofs of these can be found in [McC ] or [BMc ]. Lemma 3.2. The Tate map is the restriction of the Norm map to homotopy orbits. That is, the following diagram of weak maps commutes: Q ______//Q ____g2Gg__//_Q ` '`//`` __+___// E G E G E G E EOO | | fflfflfflffl|| ?Ø|| EhG ` ` ` ` `` ` ` ` ` ``T` ` ` ` ` `` ` ` ` ` `//EhG where T refers to the composite that defines the Tate map and the composite of the top (weak) maps is the norm map. Proposition 3.3. Let n be the n-th symmetric group. The composition Th n __t_//_T h_n___//T____//Th n induces multiplication by n! on homotopy groups. 4 The category of homotopy comonoids Let C be a pointed model category. Definition 4.1. Let X be a cofibrant object of C. We say that X is a co-H- object of C if there exists a map r : X ! X _ X which is coassociative up to homotopy, and counital up to homotopy with counit c : X ! *. 10 The co-H-objects are the comonoids of C (up to homotopy). They form a subcategory of C whose morphisms from X to Y are the maps of f 2 C which make the following diagrams commute: rX X _____//X _ X . f || f_f|| fflffl| fflffl| Y __rY_//Y _ Y Since the fold map is unital with respect to the unit u : * ! X, we can think of the fold map as a unital multiplication with which C is already equipped. The co-H-objects of C are exactly those whose comultiplication maps r act like algebra maps with respect to the multiplication map defined by the fold map. In other words, for a co-H-object X the following diagram commutes up to homotopy: _nrn fi _nX _____//_n(_n)____//__n(_nX). (6) +n|| _n+n|| fflffl| rn fflffl| X ____________________//__nX If one arranges _n _n X in an n x n-array, one can think of the map ø as the transpose map. It is a reordering of the copies of X. The first examples of co-H-objects are co-H-spaces in the category of pointed topological spaces. In particular, the basepointed circle S1 is a co- H-space. The map r in this case is the pinch map which identifies the basepoint with its antipodal point. Since the circle is a co-H-space, so are all suspensions X = S1 ^ X using the map r ^ 1. For the main theorem, we will be considering a co-H-object X and a functor F : C ! S which preserves weak equivalences. If F preserves weak equivalences, then when F is applied to diagram 6 the resulting diagram still commutes up to homotopy. The following lemma will allow us to use F , crnF , Pn and DnF interchangeably with respect to diagrams which commute up to homotopy. Lemma 4.2. Let F : C ! S be a functor which preserves weak equivalences. Then crn+1F , PnF and DnF preserve weak equivalences between cofibrant objects. 11 Proof.Recall (Definition 2.1) that crn+1F (X) is the fiber of holimP(n+1)F (Øg) ! holimP0(n+1)F (Øg). We claim that if ff : A ! B is a weak equivalence then crn+1F (ff) : crn+1F (A) ! crn+1F (B) is also a weak equivalence, as long as A and B are both cofibrant. This is true because if A and B are both cofibrant, then _kff : _kA ! _kB is again a weak equivalence for all k n + 1. Thus ff induces a weak equiv- alence on each term F (Øg) of the cube defining crn+1F , and hence induces weak equivalences on holim P(n+1)F (Øg) and holim P0(n+1)F (Øg). Then the map induced on the fiber is also a weak equivalence. Each of the remaining constructions defining PnF and DnF is a homotopy construction involving only crn+1F and F . __ |__| Remark 4.3. We only require C to be a model category in order to provide us with the correct notion of "weak equivalences" in C, and co-H-objects are only cofibrant to insure that diagrams involving co-H-objects which commute up to homotopy will still commute up to homotopy after PnF or other related functors have been applied. We can also define co-H-objects in categories C which are not model categories by requiring that all of the maps involved in the definition commute up to isomorphism. Since all functors preserve isomorphisms, lemma 4.2 is then unnecessary. For example, in the category of commutative algebras over k, the co-H-objects are almost the Hopf algebras. Notice that the co-H-objects only differ from Hopf algebras because they lack the antipodal map with which Hopf algebras are equipped. 5 The Splitting Theorem We can now state the main Theorem. Theorem 5.1. If F is a functor from C to S and X is a co-H-object of C, then rationally the fiber sequence j qn DnF (X) _____//PnF (X)____//_Pn-1F (X) 12 splits on the homotopy category. Consequently, Y P1 F (X) ' DnF (X) n 0 is a rational equivalence. Without loss of generality, we may assume F is a degree n functor by replacing F with PnF . We remark that this replacement is why Lemma 4.2 is needed. The map j exists because DnF is defined to be the fiber of the map qn. We can reformulate j ([JM4 ]) in terms of cross effects. Since F is degree n, there are equivalences DnF (X) ' (crnF (X))h n and PnF (X) ' F (X). Thus, the above fiber sequence becomes j (crnF (X))h n ____//_F (X)___//_Pn-1F (X) for each n. Recall that there is a map æ : crnF (X) ! F (_ X). The map j is n induced by æ and the fold map as follows: j crnF (X) _______________________//F (_nX) (7) | ss | | ssss | | sss | | yyyys | | F (_ X) F(+)| | n h n | | o77o LLL | | jhonoooo LLLL | fflfflfflffl||ooooo LL&&Lfflffl|| j (crnF (X))h n ____________________//_F (X) The n action on F (_nX) permutes copies of X. However, the fold map F (+) : F (_nX) ____//_F (X)has F (+)(F (oe._nX)) = F (+)(F (_nX)). Hence F (+) factors through homotopy orbits. Since the map æ is n-equivariant, æ extends to a map æh n on orbits. The resulting map j is the map for which we seek to provide a splitting. 13 When X is a co-H-object, we may extend this diagram to: "j fiOF(_nrn) crnF (X) Ø_____//_F (_nX)________//_F (_n(_nX)) (8) | | |F(+n) F(_+n)| | | | n fflffl|j" fflffl| F(rn) fflffl| crnF (X)h nØ_____//_F (X)___________//F (_nX) ffi|| fflffl| ecrnF (X) where ffi is the map dual to æ (see definition 2.9). Recall that for functors whose target category is S, crnF (X) ' ecrnF (X). We wish to show that the composite ffi O F (rn) O F (+n) O æ is the norm map T 0of section 3. If this is the case, then the following diagram commutes (as a diagram of weak maps) j crnF (X) ________//_F (_nX) (9) | q|| |F(+)| fflffl| j fflffl|F(rn) crnF (X)h n _______//_F (X)_______//_F (_ X) OO n OOOOTOO | OO ffi| OO''O fflffl| ecrnF (X)h n_i__//_ecrnF (X) where T is the Tate map. We note that the lower trapezoid of this diagram also commutes. The key point is that the Tate map, extended to a map from homotopy orbits to homotopy orbits (by including the fixed points into the orbits) induces multiplication by n! on ß*, which is rationally invertible. Thus, if f and g are two maps from crnF (X)h n to ecrnF (X) such that f Oq ' g O q then since q O i O T is rationally invertible, we actually have that f ' * *g. Furthermore, the fact that q O i O T ' n! provides the splitting. Using the square diagram 6 from section 4 as a central square, we can 14 further extend the diagram to crncrnF7(X)N7 ''ooooo NNN oooo crn(p)||NNNNN oooo fflffl| NNNN cr F (X) ______//_crnF (_ X) NN''øNNN pp7n7 n NNN = ppp | | NNN ppp j A j C NNN ppp | | NNN pp j fflffl| fflffl| crn(fNN&&fi) crnF (X) ________//_F (_nX)fiOF(_rn)//_F (_n_nX)ffi//_ecrnF_(_nX)//_ecrnecrnF* * (X) | n | qqqq | F(+n)| F(_+n) B | qqq | | n | qqq''+q fflffl| j fflffl|F(rn) fflffl| fflffl|xxqq crnF (X)h n ________//F (X)_________//F (_ X)___ffi//_ecrnF (X) OO n pp __ OOOTOOO | pppp ________OO______________________________________________________________* *__________ffi|=ppp _______OO''O_________________________________________________________* *______________________________fflffl|wwppp _____________________________________//_ h n ______________________________________ecrnFe(X)crnF (X) ________________________________________ _________________________| xn! ________________________________________________________* *_____________| ______________________________________________fflffl* *fflffl| ___))___________________________________________* *____ecrnF (X)h n (10) The squares A and B commute by the functoriality of crn (resp. ecrn). Let jfibe the composition of maps which makes the triangle C commute. We will show that the lifts j and j+ exist. Then, by the commutativity of the diagram, it will suffice to show that j = j+ O jfiO j is the norm map. Lemma 5.2. There exists a lift j such that the diagram crncrnF"(X)`55 ll ''lllll | llll | llllcr fflffl| nF(rn) crnF"(X)`__________//crnF"(_nX) ` | | | | fflffl|F(_rn) fflffl| F (_ X) _____n_____//_F (_ _ X) n n n commutes. Moreover, j is homotopic to a diagonal map. Proof.First, notice that by Remark 2.6 the following diagram commutes up 15 to homotopy: 0 Q cr F (X) crncrnF (X) _!'__//_Bij(n,n)n | OO j|| |u| fflffl| Q | crnF (_ X)_____//_ crnF (_ X) n Bij(n,n) n Q where is the diagonal map and u = ff2Bij(n,n)crnF (uff(1), . .,.uff(n)) is built of the maps ui of Remark 2.4. If we rearrange this diagram, one sees that this provides a lift: Q crnF (X) oo!0_ Bij(n,n)88OO ' crncrnFp(X)p ''qqqqq | pppp qqq |uO ppppjp qqqcr | wwpp nF(r) crnF (X)_______//crnF (_nX) In fact, since X is a co-H-object, the map r is counital. Therefor uiO r is homotopic to the identity on X for each 1 i n. It follows that j is homotopic to the diagonal map. We will not distinguish between the lift j and the associated map (!0)-1 O j . __ |__| Lemma 5.3. There exists a lift j+ such that the diagram F (_ _ X) _____//ecrnF (__X)__//ecrecrF (X) n n n nq n qqq F(+)|| crnF(+)| qq''qqq fflffl| fflffl||xx+qqqq F (_ X) ______//_ecrF (X) n n commutes. Moreover, j+ is homotopic to a fold map. Proof.This is formally dual to Lemma 5.2, using Lemma 2.10. The details __ of the proof are left as an exercise to the reader. |__| The map ø with Y Y ø : crnF (X) ! crnF (X) f2Bij(n,n) f2Bij(n,n) 16 is given on the factor indexed by f 2 Bij(n, n) by shuffling each of the n variables of crnF (X) by f. Call ø the twist map. Lemma 5.4. The map jfi, which is the composite 0 Q cr F (X) crncrnF (X) __!'_//_Bij(n,n)n | " ` PPPP | | PPPP | | PPP fflffl| Q |fflffl PPPP ! crnF (X) PPP''ø crnF (_nX)____'//_Hom(n,n) PPP PP PPPP - PPP''0øP PPPP j|| PPPP PPPP |fflffl PP''Q PPP''Q F (_ _ X) _______//_ ecrnF (X)___////_ ecrnF (X) n n - Hom(n,n) Bij(n,n) d ' || |' fflffl| fflffl|| cernF (_ X)________//_ecrecrF (X) n n n is the twist map. Proof.Note that here we have used Lemma 2.10 and the fact that coproducts are weakly equivalent to products in the lower corner of the diagram. We will begin by examining the map j0fi. Recall that æ : crnF (X) ! FQ(_nX) and ffi : F (_nX) ! ecrnF (X) are the usual maps. On the factor of -1 Hom(n,n)crnF (X) indexed by f, notice that ! : crnF (X) ! crnF (_nX) is given by crnF (cf(1), . .,.cf(n)) where ci : X ! _nX is the inclusion into the - i-th summand. Let cf := cf(1)_ . ._.cf(n). Then the map æ is given on the f-th factor by either of the composites -1 crnF (X)__cr___!________//crnF (_ X) nF(cf(1),...,cf(n)) n j || |j| fflffl| fflffl| F (_ X) _______F(cf)____//_F (_ _ X) n n n - Similar computations provide a factorization of d, projected onto the g-th factor, using the map ui : _nX ! X which is the identity on the i-th 17 summand and 0 elsewhere (this is dual to ci). Let ug := ug(1)_ . ._.ug(n). We obtain a commuting diagram -1 crnF (X) __cr___!________//crnF (_ X) nF(cf(1),...,cf(n)) n j|| |j| |fflffl fflffl| F (_ X) ______F(cf)_____//_F (_ _ X)______d_______//_ecrnF (_ X) n TT T n n n TT T |F(ug) cernF(ug(1),...,ug(n))| TT T fflffl|| | TT** fflffl| F (_ X)_________d________//ecrF (X) n n (11) The key point now is to understand the composition represented by the dashed arrow. By functoriality, we need only understand the composition (ug) O (cf(1)_ . ._.cf). To understand this composition, it is easiest to introduce the labels 0 1 0 1 X1,* X1,1 _ . . ._ X1,n B C `` '' B C B _ C c B _ _ C B . C f B . . C B .. C __________//_B.. ..C B C B C @ _ A @ _ _ A Xn,* Xn,1 _ . . ._ Xn,n |`` '' | ug | | | | |fflffl X*,1 _ . . ._ X*,n keeping in mind that X*,*= X for all choices of *'s. On any summand, Xi,*, we have cf(i) ug(f(i)) Xi,*_____//Xi,1_ . ._.Xi,n___//_Xg(f(i)),f(i) which is non-zero if and only if g(f(i)) = i. Since this is true for all 1 i * * n, this implies that f has an inverse and that g = f-1 . That is, f 2 Bij(n, n). 18 Furthermore, the composition of uf-1 with cf yields 0 1 X1,* B C B _ C B . C B ..C ____//_X1,f(1)_ . . ._ Xn,f(n) B C @ _ A Xn,* which is exactly the twist by the action of f. We have shown the composition is exactly non-zero when f 2 Bijso that diagram 11 actually defines Y Y jfi: crnF (X) ! ecrnF (X). Bij(n,n) Bij(n,n) By the equivariance of æ and d, we see that this extends to show that jfiis __ the twist map. |__| Now, looking at the composites of the maps of lemmas 5.2, 5.3 and 5.4 we find by lemma 3.2 that we have constructed the norm map. Thus the induced map T is indeed the Tate map which gives us the rational equivalence we sought. This concludes the proof of the Theorem 5.1. Remark 5.5. In fact, the proof of the theorem only requires the condition that there exists a maps F (X) ! F (_nX) which are compatible with F (+) : F (_nX) ! F (X) in the sense of definition 4.1. Since the structure maps r are actually induced as maps in C in most of our examples, we choose to state the theorem in terms of co-H-objects. However, one should note that the theorem could be stated more generally by adjusting definition 4.1 to accommodate this situation. Example 5.1. The following example is due to Tom Goodwillie. Let F : T op ! S be the functor F (X) = C*(X ^ X= ) where is the diagonal map : X ! X ^ X. We show that rationally the Goodwillie tower of F splits when X is a co-H-space, but not in general. First note that the cofiber sequence _____// ____//_X^X_ X X ^ X 19 induces a short exact sequence C*(X) _____//C*(X ^ X)_____//C*(X^X_.) The functor X 7! C*(X) is a homogeneous functor of degree one and the functor X 7! C*(X ^ X) is homogeneous of degree two. Since Dn (as well as Pn) preserve short exact sequences, we have that F (X) must also be a functor of degree at most two. We want to examine the fiber sequence D2F (X) ! P2F (X) ! P1F (X) The following 3 by 3 diagram with exact rows and columns captures all of the essential information for the Goodwillie tower of F (X): D2C*(X) ____//_D2C*(X ^ X) '___//_D2F (X) | | | | |' | fflffl| fflffl| fflffl| P2C*(X) _____//P2C*(X ^ X) _____//P2F (X) | | | | | | fflffl| fflffl| fflffl| D1C*(X) ____//_D1C*(X ^ X) ____//_D1F (X) The columns are exact since we have a fiber sequence D2 ! P2 ! P1 and, since all of the functors involved are reduced, P1 = D1. Since C*(X) is a homogeneous functor of degree 1, D2C*(X) = 0 and similarly, since C*(X ^ X) is a homogeneous functor of degree 2, D1C*(X ^ X) = 0. That means we have equivalences D2F (X) ' D2C*(X ^ X) = C*(X ^ X) and D1F (X) ' D1C*(X) = C*( X). Since F is of degree two, we also have that P2F (X) ' C*(X ^ X= ). That is, we have an exact sequence C*(X ^ X) _____//C*(X ^ X= ) _____//C*( X). inducing the long exact sequence . . .____//H*+1( X) |@| fflffl| H*(X ^ X) _____//H*(X ^ X= ) _______//H*( X) @|| fflffl| H*-1(X ^ X) ____//_. . . 20 on homology. If this map splits, then the map @ is zero. It is easy to find spaces for which this can not be the case. One simple example is given by S0. Also, notice that up to the suspension isomorphism the map @ can be expressed as * H*(X) _____//H*(X ^ X) . On (rational) cohomology, the map * : H*(X ^ X) ! H*(X) is the cup product. Therefore, if this map is zero then X has trivial cup products. Since spaces don't generally have trivial cup products, this generally doesn't split. However, when X is a co-H-space, the Hopf algebra H*(X) is exterior on the indecomposables, the cup products are trivial. We end this section by showing that the layers DnF (X) can be identified by the image of a certain map. Let r : F (X) ! F (X) be the composite r = F (+) O F (rr). Note that in the case X = S1, this induces an r-fold covering map of S1. Remark 5.6. The map Dn( r) := P1 ( r)|DnF(X) from DnF (X) ! DnF (X) induces multiplication by rn. To see this, recall that DnF (X) = (D(n)1crnF (X))h n by [JM4 ]. We have Q (n) (n) D(n)1crnF (_kX) ~= Hom (n,r)=rnD1 crnF (X) since D1 crnF is a multilinear functor. The following diagram describes r: D(n)1crnF (X) RRR |F*(rr) RRRRRRR fflffl|| RR))RQ (n) D(n)1crnF (_ X)__'_//_ D1 crnF (X) r Hom(n,r)=rn ll |F*(+)| llll+lll fflffl|uullll D(n)1crnF (X) Now, by arguments analogous to Lemmas 5.2 and 5.3, we have that the maps and + of the diagram are the diagonal and fold map (recall that coproducts and products are weakly equivalent). Hence, r is multiplications by rn. The result follows from the fact that r is a n equivariant map, so passes to homotopy orbits. 21 5.1 Higher Hochschild Homology Let A be a commutative algebra over a field k of characteristic zero. Let X. be any finite pointed simplicial set. We can form a simplicial k-algebra by composing the functor X. with the functor - A from the category of finite pointed sets to A-algebras which takes the set [[n]] = {0, 1, . .n.} to [[n]] A = A kn. The chain complex associated to the resulting simplicial algebra computes the Hochschild homology of A when X ' S1 and computes the n-th higher Hochschild homology when X ' Sn. Denote the homology of the chain complex associated to X. A by HHX*(A). Rational decompo- sitions of Hochschild homology have been studied extensively [L98 ], [GS87 ], [R93 ]. A rational decomposition of higher Hochschild homology which recov- ers the known decomposition of Hochschild homology has been discovered by 1^X Pirashvili [P00 ] and a rational decomposition for HHS* (A) was found by the first author, recovering the decompositions of the other cases [B ]. The goal is to compute the layers of the Goodwillie tower computing higher Hochschild homology in terms of the layers of Hochschild homology. The chain complex (S1 ^ X) A is a commutative differential graded Hopf algebra [B ]. Using the comultiplication map, consider (S1 ^ X) A as a co- H-object in the category of commutative augmented A-algebras, ACommA . Let U be the forgetful functor from the category ACommA to the category 1^X 1 A - mod of A-modules. By Theorem 5.1, HHS* = DnU((S ^ X) A). n The Goodwillie tower of the functor U has been computed [K-M ]. Let M be a cofibrant object of ACommA . Then the linear part, D1U(M), is I=I2(M) where I(M) is the augmentation ideal of M over A. The higher layers are given by DnU(M) = (I=I2(M)) nn= Sn(I=I2(M)) where Sn denotes the n- th part of the symmetric algebra. Since A is a cofibrant object in ACommA , so is Sd A and we will use this to compute the Goodwille tower. By Remark 5.6 the "r-fold cover map" r induces multiplication by rn on each layer DnU((S1 ^ X) A) and one can use this to show that the decomposition using Theorem 5.1 agrees with the decomposition of [B ] and hence also those of [P00 ], [L98 ] and [GS87 ]. Since D1U is a linear functor, it commutes with suspensions. Note that the suspension in ACommA is given by S1 - and the suspension in A-mod is given by ~Z[S1] A - where ~Z[S1] is the free module generated by S1. Therefore D1U(Sd A) = eZ[Sd] A D1U(S0 A). Since DnU(Sd A) = 22 (D1U(Sd A)) nnwe have DnU(Sd A) = (eZ[Sd] D1U(S0 A)) nn (12) = [eZ[Sdn] (D1U(S0 A)) n ] n (13) (14) where the action of n on the last line is given diagonally. On the first factor, n acts by permuting the copies of Sd. Each flip of factors Sd induces multiplication by (-1)d on homology. On the second factor, n acts by permuting the factors D1U(S0 A). Taking the orbits of this action produces the homogeneous degree n part of the symmetric algebra. Taking both of these actions together, we have ( dnSnD1U(S0 A) d is even; DnU(Sd A) = dn nD1U(S0 A) d isodd, where n is the homogeneous degree n part of the exterior algebra. Finally, if one then computes that DnU(S1 A) = n nD1U(S0 A) then we can express the layers of higher Hochschild homology in terms of Hochschild homology by realizing that (up to a sign) DnU(Sd A) is an d-fold suspension of DnU(S1 A). That is, the layers of higher Hochschild homology depend only on the layers of Hochschild homology. 6 Dual Calculus There is a version of calculus which is strictly dual to the Goodwillie calculus tower which we have been using so far. To obtain this theory, one simply replaces homotopy limits with homotopy colimits, coproducts with products, fibers with cofibers, etc. For details, see [McC02 ]. If one replaces coproducts by products and fibers by cofibers in 2.1, one obtains the dual cross effects, crnF (X). The dual cross effects can be thought of asQthe total cofibers of cubical diagrams. There is a natural map æ : F ( n X) ! crnF (X). One can use the dual cross effects to define a codegree n approximation 23 to F , and these assemble into a tower .. .OO | | | pn P nFO(X)O______//F8(X)8 rrr qn|| rrn-1rr | rrr p P n-1FO(X)O | | | .. . which is universal with respect to maps to F from codegree n functors. The n-th dual layer of F , DnF , is the cofiber of the map qn : P n-1F ! P nF . There is also a notion of dual co-cross effects analogous to the co-cross effects and obtained by replacing coproducts with productsQand limits with colimits. There is a natural map ~æ: ecrnF (X) ! F ( n X). To state the dual version of the theorem, we must describe the appropriate counterpart for co-H-objects. The following is the expected definition: Definition 6.1. A fibrant object X of C is an H-object if X is equipped with a map ~ : X x X ! X which is unital and associative up to homotopy. The unit map for ~ is given by the inclusion of the basepoint. Let be the diagonal map : X ! X x X. The following diagram commutes up to homotopy: Q Q n Q Q X ___fiO____//_ X n n n ~n|| Q~n || | fflffl| fflffl| n Q X ____________//_nX Q Q where ø is the map which transposes the entries of n nX. The H-objects in the category of pointed topological spaces are precisely the H-spaces. Theorem 6.2. If F is a functor from C to S which preserves weak equiva- lences and if X is an H-object of C, then rationally the cofiber sequence qn n j n P n-1F (X) ____//_P F (X)___//_D F (X) 24 splits in the homotopy category. Consequently, Y P 1F (X) ' DnF (X) n 0 is a rational equivalence. The proof of this dual version of our main theorem proceeds in essentially the same manner as the proof of the main theorem. Here is a sketch: We first reduce to the case where F is a codegree n functor, since if F is not codegree n we may replace F by P nF . When F is codegree n, we have equivalences P nF (X) ' F (X) and DnF (X) ' crnF (X)h n. We can then express the map j as in the following diagram: j n h F (X)|L_____________________//crF7(X)7 n | LLLL j* ppppp | | LLL ppp | | LL%% ppp | | Q h n || F( ) || F ( X) i| | t n | | ttt | | ttt | Qfflffl|yytt fflffl|| F ( X) ___________j__________//_crnF (X) n where F ( ) factors through the fixed points since it is n-fixed, and æ* extends because æ is n-equivariant. Now,Qusing the fact that X is an H-object and using the map eæ: fcrnF (X) ! F ( n X), we can expand this diagram to ecrnF (X) | j~| Qfflffl| F ( X) _F(~)__//F (X)___j_//crnF (X)h n n | | F(|)| |F(|) |i Q fflffl|Q fflffl|Q fflffl|| F ( X)F(~Ofi)//_F ( _X)j__//_crnF (X) n n n which commutes. We claim that the ö utside" map - that is, the composition æOF (~Oø)OF ( )Oæ~- is the norm map. Just as before, one can show this by 25 Q Q usingQthe equivalences crnF ( n X) ' Hom(n,n)crnF (X) and crncrnF (X) ' n ncr F (X) and by showing that the relevant maps are the diagonal, twist and fold maps. Once this is done, we can further expand our diagram to q n ecrnF (X)_____//_crFk(X)hOnkWWW WWWW | WWWWWiW OOOOOT |~j WWWWWW OOOO Qfflffl| WWWWWWOOO'' F ( X) __F(~)___//F (X)____j__//crnF (X)h n n | | |F(|) |F(|) |i Q fflffl|Q fflffl|Q fflffl|| F ( X) F(~Ofi)//_F ( X)__j___//crnF (X) n n n where q and i are the relevant quotient and inclusion maps, respectively. The map T is the Tate map, and this commutes with the rest of the diagram as before because the outside composition is the norm map. In fact, the entire diagram commutes except possibly for the map i. We wish to show that i provides a splitting map for j. In other words, we'll show that j O F (~) O eæO* * i is rationally homotopic to the identity map on crnF (X)h n. This is the same as showing that T OqOi is rationally homotopic to the identity map. However, recalling the definition of the map T , we have that T O q O i ~ N O i where N is the norm map since N actually lands in the fixed points. One can easily check that N O i is multiplication by n! which is rationally equivalent to the identity map on crnF (X)h n. Thus i provides the splitting. Remark 6.3. Define a map r := F (~) O F ( ). Then Dn( r) induced multiplication by rn. The argument for this is exactly dual to Remark 5.6. 6.1 The Poincar'e-Birkhoff-Witt Theorem. We seek to recover the rational version of the Poincar'e-Birkhoff-Witt Theo- rem as an application of Theorem 6.2. First, we recall the definitions required to state this theorem. This background can be found in e.g. [W94 ]. Let Liek be the category of Lie algebras over a field k of characteristic 0. If A is any algebra over k, A can be thought of as a Lie algebra by giving it the bracket [x, y] = xy - yx, where x, y 2 A. Denote A as a Lie algebra by A. Let G be a Lie algebra over k with bracket [ , ]. 26 Definition 6.4. The universal enveloping algebra of G is an algebra over k, U(G) with associated Lie algebra U(G) together with a morphism i : G ! U(G) which is universal with respect to algebras over k. That is, if f : G ! A is a map of Lie algebras, then there is a map g : U(G) ! A such that the following diagram commutes: G ___i_//DU(G) DD DDD |g* f DD!!Dfflffl|| A where g* is the map induced by g. One can construct U(G) as follows: Let i : G ! T (G) be the inclusion of G into its tensor algebra induced by including G into the first graded piece of T (G). Let I be the ideal of T (G) generated by the relations i([x, y]) = i(x)i(y) - i(y)i(x) where x, y 2 G. Then U(G) = T (G)=I. Let ß : T (G) ! U(G) be the quotient map and let Tm (G) = mj=0G j. One can see that U(G) inherits a grading from T (G) by setting Um (G) = ß(Tm (G)). If G is free as a module over k, then the Poincar'e-Birkhoff-Witt theorem shows that Um (G)=Um-1 (G) ~=Sm as k-modules, where Sm := G m = m is the m-th homogeneous graded piece of the symmetric algebra, and that U(G) ~=S as k-modules, where S is the whole symmetric algebra. In fact, if {ei}i2I is a basis for G as a k-module, then {ei1 . . .eim|ij 2 I; i1 . . .im ; m 1} is a basis for U(G) (where I is some indexing set). From [MM65 ] we know that U(G G) ~= U(G) U(G) and that U(G) is a cocommutative Hopf algebra. The multiplication structure map for the Hopf algebra is induced by concatenation on T (G), and the comultiplication is induced by the diagonal map : G ! G G. Let C be the category of cocommutative, coaugmented coalgebras over k. In C, the product is given by and the diagonal map for any object is given by the comultipication. The Hopf algebra structure makes U(G) an H-object in the category C. 27 Let F be the forgetful functor from C to the category of k-modules. We can now apply theorem 6.2 to F (U(G)) to obtain a splitting of U(G) as a k- module. We want to show that this splitting recovers the Poincar'e-Birkhoff- Witt theorem. In other words, we want to show that DnF (U(G)) ~=Sn. Note that for g 2 U1(G)=U0(G), the image of the comultiplication map is (g) = g 1 + 1 g. Denote an element in Un(G)=Un-1(G) by (g1, . .,.gn). By induction, we have X (g1, . .,.gn) = (gff(1), . .,.gff(p)) (gff(p+1), . .,.gff(p+q* *)) p+q=n ff2(p,q)-shuffles where oe, a (p, q)-shuffle means that oe is a permutation of {1, . .,.n} with oe(1) . . . oe(p) and oe(p + 1) . . . oe(p + q) Since multiplication is induced by concatenation, the map 2 = ~ O is X ~ O (g1, . .,.gn) = (gff(1), . .,.gff(p), gff(p+1), . .,.gff(p+* *q)). p+q=n ff2(p,q)-shuffles However, in U(G) = T (G)=I, for any permutation oe 2 n we have (gff(1), . .,.gff(n)) = (g1, . .,.gn). Therefore ~ O is simply multiplication by the number of ways of shuffling (g1, . .,.gn) into two factors. An easy inductive argument shows that the number of (p, q)-shuffles with p + q = n is 2n. We now know that the map 2 = ~ O induces multiplication by 2n on Un(G)=Un-1(G) ~=Sn. However we also know that 2 induces multiplication by 2n on DnF (U(G)). This shows that the two decompositions are the same - the map 2 plays the role of a linear operator on the k-vector space U(G) whose image determines the decomposition associated to it. Hence theorem 6.2 recovers the Poincar'e-Birkhoff-Witt theorem. 7 Splittings and Tate Cohomology We want to consider another extension to Theorem 5.1. 28 Definition 7.1. [McC02 ] Let F be a homotopy functor from C ! S. Define the n-th Tate cohomology of F at X to be T aten(F ; X) := cofiber((D(n)1crnF (X))h n ! (D(n)1crnF (X))h n) where the map from the homotopy orbits to the homotopy fixed points is the Tate map. Theorem 7.2. Let F be a homotopy functor from C ! S and let X be a co-H-object of C. If the map r is cocommutative, then the fiber sequence DnF (X) ! PnF (X) ! Pn-1F (X) splits whenever the Tate cohomology vanishes for all n. Proof.The proof only requires a small adjustment to the proof of theorem 5.1. If r is cocommutative (hence rn is cocommutative), then it is a n-fixed map and since ffi is n-equivariant we have a factorization F(rn) F (X) ________________________//NF (_nX) | NNNN rrr99 | | NNNN rrr | | NNN rrr | | && r | ff|| F (_ X)h n |ffi| | pp n | | ppp | | pppp h n | fflffl|wffiwpp fflffl| cernF (X)h n_____________________//ecrnF (X) This extends Diagram 5 to: j crnF (X) ________//_F (_nX) (15) | | |F(+) | | fflffl| j fflffl|F(rn) crnF (X)h n _______//_F (X)_______//_F (_ X) OO n OOOO'OO | | OO ff| ffi| OO''Offlffl| fflffl| ecrnF (X)h n____//_ecrnF (X) 29 We wish to show that oe O j is a weak equivalence. When the n-th Tate cohomology vanishes, the map T is an equivalence so that crnF (X)h n ' crnF (X)h n which in turn is equivalent to ecrnF (X)h n. Let F and G be functors to spectra. From the construction of PnF , we know that if crnF (X) ' crnG(X) via a natural transformation ! : F ! G, then there is a pull back diagram: Pn(!) PnF (X) ______//_PnG(X) (16) | | | | fflffl|Pn-1(!) fflffl| Pn-1F (X) ____//_Pn-1G(X) In this case, we wish to use G(X) = ecrnF (X)h n and ! = oe. By us- ing Lemma 5.2, weQknow that crn(ffi) O crnF (rn) is the diagonal map into ecrnecrnF (X) ' n ecrnF (X). Here, we are making use of the equivalence ecrnF (X) ' crnF (X) repeatedly. From there, one sees that Y h n cernF (X) ' ecrnF (X) n so that the map ecrn(oe) is an equivalence. Thus, we have a pullback diagram Pn(ff) PnF (X) ____________//_Pn(cernF (X)h n) | | | | fflffl| Pn-1(ff) fflffl| Pn-1F (X) __________//_Pn-1(cernF (X)h n) Now, since the Tate map is an equivalance, and since crnF (X)h n is a homo- geneous degree n functor, we have that crnF (X)h n is also a homogeneous degree n functor. So, this pullback diagram is actually F (X) ____ff//_crnF (X)h n | | | | fflffl| fflffl| Pn-1F (X) __________//* __ and the result follows by taking parallel fibers. |__| 30 Remark 7.3. Note that the theorem only actually requires that the map F (X) ! F (_nX) be öc commutative", i.e. that this map is fixed under the action of n. In [McC02 ], McCarthy proves a similar theorem for "stable functors", that is, for functors to spectra with the property that the inclusion map X _ Y ! X x Y induces an equivalence F (X _ Y ) ! F (X x Y ) for all X and Y in C. If F is a stable functor, then every object X 2 C becomes a co-H-object via the diagonal map F (X) ! F (X x X) ' F (X _ X). Since the diagonal map is cocommutative, this recovers McCarthy's theorem. In particular, functors from the category C of left (resp. right) k-modules are stable functors since it is already the case in C that products are equival* *ent to coproducts. Note also in this case that every object is both a co-H-object and an H-object, so that both the dual towers and the regular towers split when the Tate groups vanish. McCarthy's result shows that in fact the layers of the dual tower and the regular tower, and hence the two splittings, agree. 8 Appendix Recall from section 3 that if a finite group G acts on a k-module M, there is a natural map t : MG ! MG factoring the norm map as: __t0__// M MOO p|| |i| fflffl|t ?Ø| MG ____//_MG which is an equivalence whenever the order of G is invertible. We wish to extend this map to the category S of HQ-modules. Let T be an FSP. Let E_be it's associated spectrum E_k = hocolimn nE(Sn+k ) and suppose E_ is an HQ-module with an action of a finite group G. We wish to explicitly construct a weak map E_hG ! E_hG which will be the 31 correct analogue for the Tate map for modules. Such constructions can be found in e.g. [WW ], [GM ], [DGM ], [McC ]. The construction we provide h* *ere is due to Tom Goodwillie and closely follows the recollection provided in the appendix of [McC ]. We begin with some background. 8.1 Group Actions Let T op be the category of base pointed topological spaces. In the following, G is a finite group. The definitions and constructions which follow hold for all groups G, but since we are only interested in the case G = n, it won't hurt for us to assume that G is a finite group throughout. An action of a group G on a space X is a map G+ ^ X ! X. We denote the image of g ^ x by gx. Equivalently, a group action is a functor X : G ! T op where G is the category with one object and Hom G(*, *) = G. In this case, X denotes both the functor and the topological space X which is the image of the unique object of G. We say that G acts freely on X if for all non-basepoint elements x 2 X, gx 6= x unless g is the identity element of G. The following is a small collection of constructions involving G actions. The orbits of the G action on X are defined to be XG = colimGX = X={x ~ gx}. If X is a G-space and Y is a Gop-space (the action of G is on the right), then we may define a smash product Y ^ X with G-action g(y ^ x) = yg-1 ^ gx. Define Y ^G X = {yg ^ x ~ y ^ gx} and notice that (Y ^ X)G = Y ^G X. The homotopy orbits of the G action on X are XhG = hocolimG X = (EG+ ^ X)G where EG is any contractibe free G space. Later, we will use a specific model of EG given by the simplicial construction q+1^ EG = |[q] 7! G+ | 32 where + denotes a disjoint basepoint. The fixed points and homotopy fixed points of the G action on X are respectively XG = limGX = {x|gx = x for allg 2 G} and XhG = holimG X = Map (EG+ , X)G . If X is a G-space, there are two important free G spaces associated to X. The first is G+ ^ X with G-action given by h(g ^ x) = gh-1 ^ hx. The second is Map (G+ , X) with G-action given by h(f)(g) = hf(gh). There are equivalences (G+ ^ X)G ! X g ^ x 7! gx and X ! Map (G+ , X)G x 7! f(g) = gx. Since the action of G on G+ ^ X and Map(G+ , X) is free, there are G- equivariant equivalences (G+ ^ X)hG := (EG+ ^ G+ ^ X) ! (G+ ^ X)G and Map (G+ , X)G ! Map (EG+ , Map (G+ , X))G =: Map (G+ , X)hG. We conclude this section by defining an importantQrelationship between these two free G-spaces. Using the inclusion _G X ! G X we can produce a map Y G+ ^ X ~=_ X ! X ~=Map (G+ , X). G G Let this composition be named fl and notice that fl is given by ( x u = g fl(g ^ x)(u) = * otherwise. By the Blakers-Massey theorem, if X is k-connected then fl is (2k - 1)- connected. We will take advantage of this in the future to produce equiva- lences using fl and stabilization. 33 8.2 G-actions on Functors with Smash Product In this section, we assume a familiarity with the definition of functors with smash products and we describe the group actions on these. Readers who are not familiar with these definitions should consult e.g. [McC ]. If E is an FSP, then we denote by E_the spectrum associated to E with E_k= E(Sk). Since every E_ is naturally equivalent to an -spectrum (see [McC ] for a construction), we may assume that E_is an -spectrum. Define the homotopy orbits of E to be the FSP EhG with EhG(X) = 1 (E(X)hG). (Technically, we have to indicate why this is again an FSP.) Note that since EhG(X) = 1 (E(X)hG) = hocolimn n(E(Sn ^ X)hG) = hocolimn n(hocolim GE(Sn ^ X)), EhG is a homotopy colimit construction. Therefore, if E* is a simplicial F SP , the natural map |E*|hG ! |(E*)hG| is an equivalence since the geometric realization is also a colimit construction and öc limits commute". Let E_hG be the spectrum associated to EhG. The homotopy fixed points of E is the FSP with EhG (X) = ( 1 E(X))hG. Since homotopy fixed points are a limit construction we cannot make a similar statement relating |(E*)hG| with |E*|hG. However, there is a natural map |(E*)hG| ! |E*|hG. Let E_hG be the spectrum associated to EhG . We have FSP's (G+ ^ E)(X) = G+ ^ E(X) and Map (G+ , E)(X) = Map (G+ , E(X)). Note that the relevant homotopy orbit and homotopy fixed point constructions are given by (G+ ^ E)hG(X) = 1 [(G+ ^ E(X))hG] = hocolimn n[(G+ ^ E( nX))hG] Map(G+ , E)hG(X) = [ 1 Map (G+ , E(X))]hG = [hocolim n nMap (G+ , E( nX))]hG 34 8.3 The Tate Map We would like to produce a weak map E_hG__T__//E_hGwhich factors the (weak) norm map for spectra. That is, we'd like T to satisfy the following diagram of weak maps diagonalQE ff Q E _ E fold E_______//g2G_____//g2G_` ``//g2G_______//E_OO | | | | | | fflffl| T | E_hG ` ` ` `` ` ` ` ` `` ` ` ` ` `` ` ` `//E_hG where the vertical maps are the natural maps and the dotted lines are the weak maps (the dotted arrow in the top row is given by the inclusion of the coproduct into theQproduct and the Blakers-Massey theorem).QThe map oe is the "shuffle map" g2G g which acts on the factor of g2G X indexed by g by g. Before we construct the map T for a general FSP, we first consider the special case where the FSP is of the form G+ ^ E. In this case, we'll produce an equivalence (G+ ^ E)hG ! (G+ ^ E)hG. If X is a space, we almost have an equivalence (G+ ^ X)hG ' (G+ ^ X)hG given by assembling the maps from subsection 8.1: (G+ ^ X)hG (G+ ^ X)hG . (17) | ' || | fflffl| || (G+ ^ X)G flhG| | ~ | | = | | fflffl| ~= ' fflffl| X _________//_Map(G+ , X)G____//Map(G+ , X)hG Of course, if X is just a space, we do not know that flhG is a weak equivalence. Before venturing into the stable world to correct this, we make the following observation. Remark 8.1. The weak equivalence we have constructed between (G+ ^ X)hG and Map (G+ , X)hG has a very encouraging property. Notice that the 35 following diagram commutes: Map (G+O,OX)_______________ff_____________//_Map(G+O,OX) || || | ' ' | G+ ^ X _______//(G+ ^ X)hG_____//X____//_Map(G+ , X)hG where the vertical arrow on the left hand side is the diagonal map sending g ^ x to the constant map f(u) = gx (thisQmap is closely related toQfl). So far, we have considered the map oe := g2GQg to be a self map of G X. Here, we are using the identificationQof G X with Map (G+ , X). The image of the constant map f(u) = gx under u2G u is the map h(u) = ugx, which is exactly the image of g ^ x under the bottom row (off by a negative sign...? change the shuffle map to twist by g-1). In other words, we have shown that this forms a piece of the Tate map. Now, if E is an FSP then since the product and the coproduct in the category of spectra agree, we have that the G-equivariant map fl induces a G equivariant equivalence hocolimn n(G+ ^ E( nX)) ! hocolimn nMap (G+ , E( nX)). We would like to use this to produce the analogue of diagram 8.3 for FSP's. Note that since the homotopy orbits are a homotopy colimit construction, we have a (G-equivariant) equivalence (G+ ^E)hG(X) := hocolimn n[(G+ ^E( nX)hG] ' [hocolim n n(G+ ^E( nX))]hG. Using this, the equivalences (G+ ^ E)hG(X) _______'__________// 1 E(X) :=|| |:=| fflffl| fflffl| hocolimn n[(G+ ^ E( nX)hG] _'__//_hocolimn nE( nX) and hocolim n nE( nX) __'_//_Map(G+ , hocolimn nE( nX))hG 36 are induced by diagram 8.3, with 1 E(X) playing the role of X. Note that since G is finite, we also have a G-equivariant equivalence hocolim nMap (G+ , nE( nX)) _____//Map(G+ , hocolimn nE( nX)) . Since G acts trivially on n, the adjunction Map (G+ , nE( nX) ' nMap (G+ , E( nX) is also G-equivariant. Assembling these, we have a G-equivariant equivalence ' n n hG hG Map (G+ , 1 E(X))hG oo___[hocolimn Map (G+ , E( X))] := Map (G+ , E) (X* *). Now we can apply the equivalence induced by flhG to obtain (G+ ^ E)hG(X) ` `` ` ` ` ` ` ` `` ` ` ` ` ` ` //`(G+ ^ E)hG(X) | | | | ' || '|flhG| | | | | fflffl| fflffl| 1 E(X) _'_____//_Map(G+ , 1 E(X))hGo'o__Map (G+ , E(X))hG. These assemble into a natural equivalence of functors with stablilization with G-action (G+ ^E)hG ' (G+ ^E)hG. Call the weak map representing the com- posite of weak equivalence we've just described : (G+ ^ E)hG ` ` ` ` ``//(G+ * *^ E)hG . Lemma 8.2.QThe equivalence (G+ ^ E)hG ' (G+ ^ E)hG factors the shuffle map oe = g. G Proof.This follows from the fact that the diagram fl ff fl G+ ^ E ____'__//Map(G+ , E)_____//Map(G+O,OE)oo'_____G+ ^OXO | | | | | | fflffl| | fl | (G+ ^ E)hG __'_____// 1 E______'_//Map(G+ , E)hGo'o__(G+ ^ X)hG commutes. The rectangle on the left hand side commutes by remark 8.1 and the fact that the weak equivalences indicated are induced by weak equiv- alences of spaces. The square on the right commutes because fl is a G- __ equivariant map. |__| 37 We now have enough information to construct the Tate (weak) map. Let ß be the G-equivariant equivalence EG+ ^ E ! E. The Tate map is the composite EhGOOØ ' iØØ V q+1 (EG+ ^ E)hG ______:=__//_|[q] 7! ( G+ ^ E)|hG OOØ ' ØfØ V q+1 |[q] 7! ( Ø G+ ^ E)hG| ' Ø| | V fflfflØ |[q] 7! ( q+1G+ ^ E)hG| |g| V fflffl| oo__:=_____ |[q] 7! ( q+1G+ ^ E)|hG |EG+ ^ E|hG ' |i| fflffl| EhG where f and g are induced by the universal properties of the homotopy orbits and homotopy fixed points. Note that g is not neccessarily an equivalence. Remark 8.3. Technically, we have only defined the weak map on simplicial level 0. We can extend to a simplicial map by using the extra degeneracy, d0, with which EG is equipped (since it is the path space of BG). See for example ([W94 ], section 8.3) for details. Lemma 8.4. The Tate map is the restriction of the Norm map to homotopy orbits. That is, the following diagram of weak maps commutes: Q ______//Q ____g2Gg__//_Q oo'```` __+___// E G E G E G E EOO (18) | | fflfflfflffl|| ?Ø|| EhG ` ` ` ` `` ` ` ` ` `` t`` ` ` ` `` ` ` ` ` `//EhG where T refers to the composite that defines the Tate map. 38 Q Proof.Using the equivalences G+ ^ E ' _G E ' G E ' Map(G+ , E), we can rewrite the Norm map as ____//_ _ff_//_ +___//_ E G+ ^ E G+ ^ E E where all of the maps are homotopic to the maps in diagram 18 above, but composed with the appropriate weak equivalences. Under the identification G+ ^ E ' _G E, the fold map is the same as the projection map G+ ^ E ! E. Since the map ß : EG+ ^ E ! E is given simplicially by the projections q+1^ G+ ^ E ! E ß is the fold map in simplicial degree 0. If one identifies G+ ^ E with Map (G+ , E) instead, the diagonal map given (on the space level) by tak- ing x 2 E(X) to the constant map with value x is a homotopy inverse to the fold map G+ ^ E ! E. Therefore, ß-1 is the diagonal map in simplicial degree 0. For any space or FSP X, let X.denote the constant simplicial set whose value in every simplicial dimension is X and whose face and degeneracy maps are idX . We wish to show first that the weak map ß-1 factors the diagonal map : |E.| ! |(G+ ^ E).|Qwhich makes up the first part of the norm map (recall that G+ ^ E ' G E). To do this, we want to show that |E.|____________________________//_|(G+ ^ E).| 2FF2 22FFF | 22 FFF fflfflffl|| 22 F""F -1 22 |E | ` `` ` i`` //` q+1V 22 . hG |[q] 7! G+ ^ E|hG 22 | 22 |' ' | 22| fflffl|| ßß2fflffl| |i-1| q+1 |(EhG).|`` ` ` ` `//|[q] 7! ( V G+ ^ E)hG| commutes. By remarkV8.3, we need only worry about simplicial map involving EG+ := |[q] 7! q+1G+ | in simplicial degree 0, since such maps naturally extend to all degrees by using d0. The map ffl is the inclusion of simplicial q+1V degree 0 of |(G+ ^ E).| into |[q] 7! G+ ^ E|, extended in this way, and 39 followed by the map to homotopy orbits. The square which comprises the lower right hand corner of the diagram commutes by the equivariance of , and of ß. The triangle on the left hand side of the diagram commutes by the universal properties of the colimits involved. In simplicial degree 0, the outer diagram is _______//_ E G+ ^ E | | | | |fflffli-1 fflffl| EhG ____//_(G+ ^ E)hG which commutes because the diagonal map is a homotopy inverse to ß, and by the equivariance of ß. Once again, we extend this to a diagram of simplicial maps by using d0 and this whole diagram commutes. We already know that G+ ^ E ___ff___//_G+O^OE | | | | |fflffl | (G+ ^ E)hG `` `//(G+ ^ E)hG commutes by Lemma 8.2. We can again extend this to a commuting diagram of simplicial sets by remark 8.3. Finally, dualize the argument used with the diagonal map to show that |(G+ ^OE).|____________+_______________//_|E.|O<