p-compact groups as framed manifolds Tilman Bauer Department of Mathematics, Rm. 2-492, Massachusetts Institute of Technology, Cambridge (MA) 02139 _____________________________________________________________________________ Abstract We describe a natural way to associate to any p-compact group an element of the p-local stable stems, which, applied to the p-completion of a compact Lie g* *roup G, coincides with the element represented by the manifold G with its left-invar* *iant framing. To this end, we construct a d-dimensional sphere SG with a stable G- action for every d-dimensional p-compact group G, which generalizes the one-poi* *nt compactification of the Lie algebra of a Lie group. The homotopy class represen* *ted by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with SG . Key words: p-compact groups; Thom-Pontryagin construction; adjoint representation; transfer _____________________________________________________________________________ 1 Introduction Let G be a compact Lie group with (real) Lie algebra g = TeG. Left mul- tiplication with an element g 2 G gives an isomorphism g ~= TgG, and by choosing a basis for g, we thus obtain a framing L of the manifold G, called the left-invariant framing. The Pontryagin-Thom construction produces from this data an element in ßsd(S0), where d = dim G. Computations of homo- topy classes that arise in this way have been made by Smith [Smi74 ], Wood [Woo76 ], Knapp [Kna78 ], and others. The most extensive table of homotopy classes represented by Lie groups can be found in [Oss82 ]. This construction is intimately related to the transfer map for the universal bundle over the classifying space of the Lie group G. More generally, for every i subgroup inclusion H ,! G of Lie groups, there is a transfer map in the stable i! 1 h g homotopy category 1 BGg ! BH . Here, BG stands for the Thom space of the bundle associated to the adjoint representation of G on g. This Preprint submitted to Elsevier Science May 1, 2002 map is a twisted version of the well-known Umkehr map for the fibration G=H ! BH !i BG, BG+ ! BH , (1.1) where stands for the normal bundle along the fibers of p. Note that the tangent bundle along the fibers of p is g=h and hence = h - g as virtual vector bundles. By taking Thom spaces with respect to the bundle g resp. p*g on both sides in (1.1), we obtain the desired map. Lemma 1 The homotopy class represented by the d-dimensional compact Lie group G is given by the following composite of maps: i! 1 0 Sd ! 1 BGg ! EG+ ' S . Here the left hand map is the inclusion of the bottom cell into BGg, and i is the inclusion of the trivial subgroup into G. Note that we can factor this map through any BHh, where H < G. For H = T a maximal torus in a semisimple G, this leads to an explicit way of computing the corresponding element in ßsd. In this paper, we go one step further and show that the transfer functor (-)! can be extended to the class of all p-compact groups. A p-compact group ([DW94 ]) G is a H*(-; Z=p)-local space BG such that G =def BG has totally finite mod-p homology. Prominent examples are given by HZ=p-localizations of compact Lie groups. Dwyer and Wilkerson have worked out an extensive Lie theory of p-compact groups [DW94 ]. It turns out that the classification of p-compact groups, at least at odd primes, boils down to the classical classifi- cation of complex reflection groups by Sheppard and Todd [ST54 ], refined by Clark and Ewing [?] to p-adice reflection groups. These groups occur as "Weyl groupsö f p-compact groups, and the p-compact groups themselves have been constructed on a case-by-case basis; no general method to construct them from their Weyl groups is known so far. The main results of this paper are Theorem 2 (1) For every p-compact group G of Fp-homological dimension d, there is a HZ=p-local d-dimensional sphere SG with a stable G-action, which in the case of the localization of a compact Lie group is equivalent to the localization of the one-point compactification of the Lie algebra g with the adjoint action. (2) For every monomorphism H < G of p-compact groups, there is a map SG ! G ^H SH which is an isomorphism in Hd(-; Fp). 2 Annoyingly, the morphism in (2) fails to be G-equivariant, but it does so in a well-behaved manner. In fact, there is an extension to EG+ ^ SG ! G ^H SH that is G-equivariant. Theorem 3 There is a contravariant functor t from the category of p-compact groups and monomorphisms to the stable homotopy category with the following properties: (1) The spectrum BGg := t(G) = EG+ ^G SG is Z=p-local and connective, and H*(BGg; Zp) is a free module over H*(BG; * *Zp) on a Thom class in dimension d, the dimension of G. (2) The functor t makes the following diagram commute: SG ______//G ^H SG | | | | |fflffl |fflffl BGg _______//BHh (3) The composition t O Lp, defined on the category of compact Lie groups and monomorphisms (where Lp is HZ=p localization), is equivalent to the functor Lp O (-)!. A few explanations are in order. A monomorphism of p-compact groups is, by f definition, a pointed map BH ! BG whose homotopy fiber has finite mod p homology. Hence, even for Lie groups, we allow additional maps such as _k x unstable Adams operations LpBU(n) ! LpBU(n) (k 2 Zp ). Of course, in that case the map is a homotopy equivalence and will not yield an interesting transfer map. Theorem 3 enables us, by means of Lemma 1, to associate to any p-compact group an element in the stable stems, which one might provocatively call "the p-compact group in its invariant framing". Table 1 shows p-compact groups with the homotopy classes they represent. Notation. The symbol Zp denotes the p-adic integers. All homology and coho- mology theories in this paper are assumed to be reduced, and all spaces to be compactly generated weak Hausdorff. 3 _________________________________________________________________________ | Name | dim | rank | ST number | prime |homotopy class | |__________|_________|______|___________|___________|____________________|_ | An |n(n + 2) | n | 1 |any | , ... | | | | | | | | | X(m, q, n) | * | n | 2a |1 (m) |? | | | | | | | | | I2m |2m + 2 | 2 | 2b | 1 (m) |0 | | | | | | | | | ~m |2m - 1 | 1 | 3 |1 (m) |ff1 for m = p - 1 | |____________|_______|______|____________|__________|____________________ | | | 18 | 2 | 4 |1 (3) |0 | | | | | | | | | | 34 | 2 | 5 |1 (3) |0 | | | | | | | | | | 30 | 2 | 6 |1 (12) |0 | | | | | | | | | | 46 | 2 | 7 |1 (12) |0 | | | | | | | | | | 38 | 2 | 8 |1 (4) |fi1 for p = 5 | | | | | | | | | | 62 | 2 | 9 |1 (8) |0 | | | | | | | | | | 70 | 2 | 10 |1 (12) |0 | | | | | | | | | | 94 | 2 | 11 |1 (24) |0 | | | | | | | | | Za2 | 26 | 2 | 12 |1, 3 (8) |0 y | | | | | | | | | | 38 | 2 | 13 |1 (8) |0 | | | | | | | | | | 58 | 2 | 14 |1, 19 (24) |0 | | | | | | | | | | 70 | 2 | 15 |1 (24) |0 | | | | | | | | | | 98 | 2 | 16 |1 (5) |0 | | | | | | | | | | 158 | 2 | 17 |1 (20) |0 | | | | | | | | | | 178 | 2 | 18 |1 (15) |0 | | | | | | | | | | 238 | 2 | 19 |1 (60) |0 | | | | | | | | | | 82 | 2 | 20 |1, 4 (15) |0 | | | | | | | | | | 142 | 2 | 21 |1, 49 (60) |0 | | | | | | | | | | 62 | 2 | 22 |1, 9 (20) |0 | |____________|________|_____|____________|___________|___________________ | | | 33 | 3 | 23 |1, 4 (5) |0 | | | | | | | | | DW3 | 45 | 3 | 24 |1, 2, 4 (7)w|for p = 2? | | | | | | | | | | 51 | 3 | 25 |1 (3) |0 | | | | | | | | | | 69 | 3 | 26 |1 (3) |0 | | | | | | | | | | 93 | 3 | 27 |1, 4 (15) |0 | |____________|________|_____|____________|___________|___________________ | | F4 | 52 | 4 | 28 |any |? | | | | | | | | | Za4 | 84 | 4 | 29 |1 (4) |0 | | | | | | | | | | 124 | 4 | 30 |1, 4 (5) |0 | | | | | | | | | | 124 | 4 | 31 |1 (4) |fi1fi2 for p = 5? | | | | | | | | | | 164 | 4 | 32 4 |1 (3) |fi2 for p = 7? | |____________|________|_____|____________|__________|__1_________________ | | | 95 | 5 | 33 |1 (3) |0 | |____________|________|_____|____________|__________|____________________| | Ag6 | 258 | 6 | 34 |1 (3) |0 | | | | | | | | | E6 | 78 | 6 | 35 |any |fi3 for p = 3? p = 2? | |____________|________|_____|____________|__________|__2_________________ | | E7 | 133 | 7 | 36 |any |? | |____________|________|_____|____________|__________|____________________| | E8 | 248 | 8 | 37 |any |? | |____________|________|_____|____________|__________|____________________| * m(n2 - n + 2n_ q ) - n y does not vanish for purely dimensional and filtration reasons. Table 1.1 p-compact groups and the homotopy classes they represent. 2 HZ=p-local equivariant spectra 2.1 HZ=p-localization and p-completion In [Bou79 ], a localization functor X ! XE is constructed for every spectrum E with the property that X ! XE is the terminal E*-equivalence out of X. We will need this for E = HZ=p, the Eilenberg-MacLane spectrum with coefficients in Z=p. If X is connective, this functor is very well-behaved: Lemma 4 (Bousfield [Bou79 ]) Let X be a connective spectrum. Then lo- calization with respect to HZ=p is equivalent to localization with respect to M(Z=p), the Moore spectrum for Z=p. This localization can be constructed explicitly as the p-completion of X, i.e. n o XM(Z=p) = X^p= holim--.-.!.X ^ M(Z=p3) ! X ^ M(Z=p2) ! X ^ M(Z=p) . I will denote the HZ=p-localization functor by Lp. For a finite spectrum X, smashing with X commutes with homotopy limits, and therefore LpX = X ^ LpS0. Let S be the full subcategory of HZ=p-local spectra, i.e. of spectra X such that X ! LpX is a weak equivalence. This category has all homotopy limits, homotopy colimits, smash products and function spectra if we compose the usual construction with the functor Lp. (In fact, a homotopy limit of E-local spectra is already E-local.) The smash product is associative up to homotopy, with unit object LpS0. When working in S, I will omit any mention of Lp and also write S0 for the unit of the smash product. 2.2 G-spectra To construct the transfer map t, we will need to work in a point-set category of equivariant spectra. For our purposes, it is enough to work in the category of so-called naive G-spectra. I will drop the word än ive" since it will make this work appear so puny. Let GS be the category whose objects are HZ=p-local spectra E, together with a (left) G-action on every space En (n 2 Z), such that the structure maps En ! En+1 are G-equivariant homeomorphisms. Morphisms are defined as usual. This category has again all homotopy limits and colimits, smash products, and function spectra. The unit is given by LpS0 with the trivial G-action. It may be worth pointing out that the G-action on 5 a smash product is the diagonal one, whereas the G-action on map (X, Y ) is given by conjugation. There are at least two notions of equivariant equivalences in GS, and it is important to distinguish between them. Definition 5 I will call a G-equivariant map f : X ! Y between G-spectra a coarse G-equivalence if it is a weak equivalence of underlying spectra. It is called a G-homotopy equivalence if there is an inverse map up to homotopies through G-equivariant maps. For a Lie group G, a coarse equivalence f that also induces an equivalence on H-fixed points for every closed subgroup H is sometimes called a weak G-equivalence. By the equivariant Whitehead theorem for spaces with for a Lie group action of G (cf. [Ada84 ], [LMS86 ]), a weak G-equivalence between G-CW complexes is a G-homotopy equivalence; this need not be true for coarse G-equivalences in general. For example, the obvious coarse G-equivalence EG ! * does not have an equivariant inverse. Define a free G-CW spectrum to be a G-spectrum which is built from cells of the form Sn ^ G+ . Lemma 6 If E is a free G-CW spectrum and X ! Y is a coarse G-equivalence of G-spectra, then it induces weak equivalences map G(E, X) ~! map G(E, Y ) and E ^G X !~ E ^G Y. PROOF. Both equivalences are clear if E is a single cell Sn ^ G+ because in that case, map G(E, -) = map (Sn, -) and E ^G X = Sn ^ X. It follows for finite spectra by induction and the five-lemma, and in general by a direct limit argument. 2 For a G-spectrum X, define XhoG = EG+ ^G X = (EG+ ^ X)=G and XhoG = map G (EG+ , X) where map G denotes G-equivariant based maps. 6 The spectrum 1 EG+ is a free G-CW spectrum. Therefore Lemma 6 implies in particular that a coarse G-equivalence f : X ! Y induces weak equivalences fhoH : XhoH ! Y hoH and fhoH : XhoH ! YhoH for any subgroup H < G. If H is normal in G then these maps are coarse G=H-equivalences. 2.3 Duality For a nonequivariant spectrum X, let DX =defmap (X, S) be its dual. This spectrum DX will not have good duality properties in general. For instance, there is no guarantee that D(DX) ' X. We call X strongly dualizable if there '' is a map S ! X ^ DX such that the following diagram commutes up to homotopy: ______''__// S X ^ DX (2.1) |'| |fi| fflffl| fflffl| map (X, X) oo___DX ^ X Here, ø is the flip involution, ' is adjoint to the identity map X ! X, and is the map adjoint to X ^ DX ^ X eval^idX-!X. The existence of such a map j is equivalent to being a homotopy equivalence. It implies that D(DX) ' X. Cf. [May96 ]. It turns out that the category GS contains very few strongly dualizable ob- jects, i.e. objects for which in the above diagram, there is an equivariant map j, or equivalently, is a G-homotopy equivalence. This is mainly due to the fact that we are considering naive G-spectra. For example, if M is a com- pact G-manifold, we usually construct a duality morphism j by embedding M equivariantly into some G-representation V , use the Thom-Pontryagin- construction to get an equivariant map SV ! M ^ M+ , and desuspend by SV . This last step is impossible in the category of naive G-spectra unless V is a trivial representation, i. e., unless M has a trivial G-action. If G is a Lie group, and we work in the category of non-naive G-spectra, it is known that a G-CW spectrum is strongly dualizable if and only if it is a wedge summand of a finite G-CW spectrum. It seems plausible that if one succeeded to set up the "right" category of non-naive G-equivariant spectra for a p-compact group G, all the objects in this work that are nonequivariantly dualizable but do not appear to be strongly dualizable in GS would actually have a strong dual in that category. From a philosophical point of view, this 7 would be desirable and make some cumbersome technical problems disappear. However, in my opinion, the effort needed for setting up such a category is not warranted by the purposes of the present work. Suppose that X is a G-spectrum that, as a nonequivariant spectrum, is strongly dualizable. Then the map : DX ^ X ! map (X, X), which is always G- equivariant by naturality, is a coarse G-equivalence, and j exists but is not necessarily G-equivariant. As should be expected, X will have about half of all the good properties of a strongly dualizable object. For instance, there is a weak equivalence map G (A, B ^ DX) ! map G (A ^ X, B) (2.2) given by A ^ X ! B ^ DX ^ X id^eval-!B but in general no such map map G(A ^ DX, B) 6! map G (A, X ^ B). A spectrum or space X is called p-finite if H*(X; Fp) is totally finite. The following lemma has a rather long history of my advisor suggesting a proof using the Adams spectral sequence and me rejecting it and finding another (erroneous) proof without it. Eventually, I caved in. Here's his proof. Kudos for Mike. Lemma 7 For every connective, HZ=p-local, p-finite spectrum X, there is a finite spectrum X0 and p-equivalence X0 ! X. Remark. This association is not claimed to be functorial. PROOF. Let k 2 Z be minimal with Hk(X; Fp) 6= 0. We proceed by induc- tion on the size of H*(X; Fp). We will first show that there is a nontrivial map f : ßk(X) ! Hk(X) ! Hk(X; Fp). This would be a simple application of the Hurewicz theorem relative to a Serre class if the class of groups that vanish when tensored with Fp were actually a Serre class, which it is not. Since X is connective and HZ=p-local, its HZ=p-nilpotent completion and its HZ=p-localization agree (Lemma 4) and are equal to X, hence the classical 8 s,t HZ=p-based Adams spectral sequence converges to ß*(X). Since ExtA* (H*(X; Fp), * *Fp) = 0 for t - s < k, the Hurewicz map f : ßk(X) ! Hk(X; Fp) has to be nonzero. Let fi : Sk ! X be a map such that f([fi]) 6= 0. Let F be the HZ=p-localization of the homotopy fiber of fi. F is p-finite, HZ=p-local and the size of its Z=p- homology is smaller than that of X, hence by induction, there is a finite spectrum F 0and a p-equivalence F 0! F . Let X0 be the cofiber of F 0! F ! Sk; X0 is a finite spectrum and comes with a map X0 ! X which is a p-equivalence. 2 Corollary 8 Let X be a connective, HZ=p-local, p-finite spectrum. Then X has a strong dual in S. PROOF. By Lemma 7, there is a finite spectrum X0 and a p-equivalence X0 ! X. Hence there is a p-equivalence of HZ=p-local spectra LpX0 ! LpX = X, which therefore is a weak equivalence. It remains to show that Lp(D(X0)) is a strong dual of LpX0 for a finite spectrum X0. We need to show that Lp(map (X0, S)) = Lp map (LpX0, LpS). Indeed, Lp map (LpX0, LpS) ' map (X0, LpS) ' DX0^ LpS ' Lp(DX0). Now j : S ! X ^ DX induces a duality map Lpj : LpS ! Lp(X0^ DX0) = LpX0^ LpDX0 = LpX0^ D(LpX0), which shows that Lp(D(X0)) is a strong dual. 2 3 p-compact groups This section will provide some background about p-compact groups, a topic that has become very popular starting in the early nineties, largely due to some beautiful work of Dwyer and Wilkerson [DW94 ] and Dwyer, Miller, and Wilkerson [?]. 9 3.1 Definition and examples Definition 9 ([DW94 ]) A p-compact group is a triple (X, BX, e) where BX is a HZ=p-local space, X is an Fp-finite space, and e : X ! BX is a homotopy equivalence. As noted in the introduction, the HZ=p-localization LpG of a Lie group G gives rise to a p-compact group (LpG, Lpe, LpBG) for every prime p. Here e : G ! BG is the canonical equivalence. To illustrate how to obtain other p-compact groups, it is instructive to recall the connection between spaces with polynomial cohomology rings and finite loop spaces. If X is a space such that H*(X; Fp) ~=Fp[oe1, oe2, . .,.oer] with oei 2 Hdi(X; Fp), di even, then by the Eilenberg-Moore spectral sequence, ^ H*( X; Fp) ~= {ø1, ø2, . .,.ør} with øi 2 Hdi-1( X; Fp), and øi is the image of oei under the transgression. In particular, H*( X; Fp) is finite, and LpX is a p-compact group. The reader should be warned that not all p-compact groups are polynomial in this sense. A large class of p-compact groups, called the non-modular groups, can be constructed as follows: First pick a finite group W < GLr(Zp) (a "Weyl" group for the p-compact group); W acts on Zrpand hence also on K(Zrp, 2) = Lp(CP 1)r. Define a space i j BG =defLp K(Zrp, 2)hoW . We want to determine what restrictions on W we have to make to ensure that BG is a space with polynomial cohomology. There is a spectral sequence converging to H*(BG; Fp) whose E2 term is Er,s2= Hr(BW ; Hs(K(Zp, 2); Fp)) = Hr(BW ; Fp[[t1, . .,.tr]]) If p does not divide |W |then Er,s2= 0 for r > 0, and E0,s2= Fp[[t1, . .,.tr]]W = Hs(BG; Fp) Theorem 10 (Sheppard-Todd, Clark-Ewing [ST54 ,?]) Let W < GLr(Fp) be finite. If Fp[t1, . .,.tr]W is polynomial then W is a pseudo-reflection group, i.e., i* *t is generated by a finite set of finite order elements that fix a hyperplane in Frp. 10 The converse is true if (but not only if) p does not divide the order of W . 2 Moreover, in the non-modular case, every representation of W over Fp can be lifted to a representation over Zp. We can thus construct a p-compact group BG for every pseudo-reflection group defined as a subgroup of GLr(Zp) such that p does not divide the order of W . All such groups are classified [ST54 ,?], and Table 1 lists some statist* *ics about them. In that table, all exotic groups of rank bigger than 1 that are given a name are non-modular. Definition 11 A morphism BH ! BG of p-compact groups is just a pointed map Bf : BH ! BG. It is a monomorphism if its homotopy fiber is Fp- finite, and an epimorphism if its homotopy fiber is a p-compact group. Two morphisms BH ! BG are called conjugate if they are freely homotopic. For Lie groups H and G, being conjugate in the p-compact sense is indeed the same as being conjugate as Lie group homomorphisms. 3.2 Maximal tori In the non-modular case considered in the previous subsection, BG naturally comes with a map BT := K(Zrp, 2) ! Lp(K(Zrp, 2)hoW ) = BG given by the inclusion of the fiber of the bundle BG ! BW . Call a monomor- phism of p-compact groups BT ! BG a maximal torus if BT = K(Zrp, 2) for some r, and it does not factor through a larger torus. One of the main results of [DW94 ] is that such tori also exist in the non-modular case: Theorem 12 (Dwyer-Wilkerson [DW94 ]) (1) For every connected p-compact group BG, there is a maximal torus BT ! BG, unique up to conjugacy. (2) The monoid map BG (BT, BT ) of endomorphisms of BT over BG is homo- topy equivalent to a finite group W acting as a group of pseudo-reflections on H2(BT ; Zp) ~=Zrp. (3) H*Qp(BG+ ) ~=H*Qp(BT+ )W , and H*Qp(BT+ ) is a free module over H*Qp(BG+ ). 2 11 Here, H*Qp(X) =def H*(X; Zp) Zp Qp. (Note that H*(X; Qp) would be an unreasonably large group; whereas Hom (Zp, Zp) = Zp, we have Hom (Zp, Qp) = Hom (Qp, Qp) = Q@2p.) Corollary 13 The p-compact flag variety G=T = hofib(BT ! BG) has H*Qp(G=T+ ) = H*Qp(BT+ ) = (H*Qp(BT )W ). PROOF. There is an Eilenberg-Moore spectral sequence s,t * * Es,t2= dTorH*Q(BG )(HQ (BT+ ), Qp) =) HQ (G=T+ ), p + p p where Tdorsis the sth derived functor of the completed tensor product ^ . In this spectral sequence, Es,t2= 0 for s > 0 because H*Qp(BT+ ) is free, hence flat, over H*Qp(BG+ ), and E0,t2= H*Qp(BT+ ) ^H*Qp(BG+)Qp = H*Qp(BT+ ) = (H*Qp(BT )W ). 2It will become important in calculations to know exactly what the degree of the map c : H*(BT+ ; Zp) = (H*(BG; Zp)) ,! H*(G=T+ ; Zp) is in the top dimension. Lemma 14 Let p > 2 or G of Lie type or G = DW3. Then the cohomology ring H*(G=T ; Zp) is concentrated in even dimensions and torsion free. PROOF. This is a result that follows from Schubert calculus in the case where G is a Lie group. For polynomial p-compact groups, we have H*(G=T+ ; Zp) ~=H*(BT+ ; Zp) = (H*(BG; Zp)) by the same argument as in Corollary 13, and the assertion holds. Now the only non-polynomial p-compact groups for odd p are [KM97 ,?,?]: o Type An with a fundamental group that is a p-group; o types F4, E6, E7, E8 for p = 3; and o type E8 for p = 5. In particular, they are all Lie groups. 2 12 Remark 1. Since the classification of 2-compact groups in not finished at this time, we cannot claim Lemma 14 holds for p = 2. However, the only known non-Lie 2-compact group is DW3, which is polynomial [DW93 ]. It is conjec- tured that it actually is the only one. Remark 2. It would be much more satisfying to find a proof that does not rely on the accidental fact that all non-polynomial p-compact groups are of Lie type. For example, it would be exciting to produce a Schubert calculus for p-compact flag varieties. I am grateful to Nitu Kitchloo for pointing out to me the implication (i)) (iii) in the following proposition: Proposition 15 Let G, p be as in Lemma 14 and G be simply connected. Then the following are equivalent: (i) c is an isomorphism in the top dimension; (ii) c is an isomorphism in all dimensions; (iii)H*(G=T+ ; Zp) is generated by degree 2 classes; (iv) H*(BG+ ; Zp) has no torsion; (v) H*(BG+ ; Zp) is a polynomial algebra. PROOF. If G is simply connected it follows from the Serre spectral sequence associated to G=T ! BT ! BG that ~= 2 H2(BT+ ; Zp) -! H (G=T+ ; Zp). This shows (iii), (ii)) (i). For (iv)) (ii), assume c fails to be an isomor- phism in dimension k. Then in the above Serre spectral sequence, a class x in Hk(G=T+ ; Zp) has to support a nontrivial differential di. Since rationally, H*Qp(G=T+ ) is always generated by degree 2 classes, di(x) has to be a torsion class in Hi(BG+ ; Hk+1-i(G=T+ ; Zp)). By Lemma 14, the latter group is isomorphic to Hi(BG+ ; Zp) ^Hk+1-i(G=T+ ; Zp). Since by the same lemma, H*(G=T+ ; Zp) is torsion free, there must be a torsion class in Hi(BG+ ; Zp). For (i)) (iv), assume y 2 Hj(BG+ ; Zp) is torsion with j minimal. By the multiplicativity and Lemma 14, this implies that y = dj(x) for some x 2 Hj-1(G=T+ ; Zp). Pick a generator g 2 Htop(G=T ; Zp). Now 0 = dj(gx) = dj(g)x gdj(x). 13 Since dj(x)g = yg 6= 0, di(g) cannot be trivial, hence g is not a permanent cycle, and c is not an isomorphism in the top dimension. For (iv), (v), note that if H*(BG+ ; Zp) has no torsion, it has to be concen- trated in even degrees since it injects into H*Qp(BG+ ). Hence H*(G+ ; Zp) is a degreewise free Hopf algebra on odd-dimensional generators, which implies that it is an exterior algebra. Hence, by the Serre spectral sequence for the path-loop-fibration on BG, H*(BG+ ; Zp) is a polynomial algebra. 2 3.3 A comment on rigidity In the definition of a p-compact group (X, BX, e), the data X and e are redundant and probably only classically included to provide some justification for speaking of ä p-compact group Xä nd not the more accurate "BX". On the other hand, it is always possible to choose a model for the loop space X := BX such that X is actually a topological group and not just an H- space. A possible construction is the geometric realization of Kan's loop group functor G as described in [Kan56 ]. Let S denote the category of simplicial sets and S0 the full subcategory of reduced simplicial sets, i.e., simplicial sets X such that X0 = pt. Equip S0 with the projective model structure, i.e. weak equivalences and cofibrations are shared with S. It turns out ([GJ99 ]) that a map X ! Y between fibrant reduced simplicial sets is a fibration if and only if it induces a surjection on fundamental groups. Let s Gr denote the category of simplicial groups, carrying the injective model structure (sharing weak equivalences and fibrations with the underlying sim- plicial sets). Proposition 16 (Kan) There is a Quillen equivalence ___ W : s Gr $ S0 : G Furthermore, there is a Quillen equivalence between the category S0 and the category of connected, pointed simplicial sets, Sc, where the functor F : Sc ! S0 is given by F (X)n = {x 2 Xn | i*(x) = * for everyi : [0] ! [n]}. 14 Passing to topological spaces, we also have Quillen equivalences Sc $ {pointed connected topological spaces } and s Gr $ {topological groups} This suggests the following alternative definition of a p-compact group: Definition 17 (alternative) The category of p-compact groups is the full subcategory of all HZ=p-local topological groups (compactly generated, weak Hausdorff) whose objects are fibrant, cofibrant, Fp-finite, and such that ß0(G) is a finite p-group. The condition on the group of components is necessary to ensure that BG is still HZ=p-local. By the above Quillen equivalences, every map BH ! BG is, up to homotopy, induced by a group homomorphism H ! G if H and G are p-compact groups in this sense. Moreover, a monomorphism BH ! BG in the sense of the original definition is always, up to homotopy, induced by an injective group homomorphism H ! G. In fact, we can functorially replace BH ! BG by a cofibration, and Kan's functor G preserves cofibrations. Cofibrations of simplicial groups are injective. We will therefore work in the category of p-compact groups according to the above alternative definition, and define monomorphisms as actual subgroup inclusions. 4 Adjoint representations Although much of Lie theory carries over to the more general setting of p- compact groups, the representation theory, and in particular the adjoint rep- resentation, does not seem to have a direct analogue for p-compact groups. We do not know how to construct a vector bundle on a p-compact group BG that plays that role, but we can manufacture something that, in the Lie cases, looks like its Thom spectrum. Definition 18 For any connected p-compact group G, define op SG = ( 1 G+ )hoG . Note that G acts on 1 G+ by both left and right multiplication. We agree to use the right action for the formation of this homotopy fixed point spectrum, 15 leaving us a left G-action on SG . The adjoint Thom spectrum of G is the spectrum BGg =def(SG )hoG = EG+ ^G SG . Klein [Kle01 ] has shown that this construction for G a (non-localized) con- nected compact Lie group indeed gives rise to the Thom spectrum of the adjoint bundle. It is therefore reasonable to mimick this construction for a p-compact group G. The main point of this section is to show that SG , defined as above for a connected p-compact group G, is homotopy equivalent to a sphere. We will need two classical lemmas on finite-dimensional Hopf algebras. All cohomology and homology groups are with coefficients in Fp. Lemma 19 If G is a topological group such that H*(G) is totally finite, then H*( 1 G+ ) is a free H*(G)-module on a generator in dimension dim G. PROOF. Note that A = H*(G) is a Hopf algebra, and H*( 1 G+ ) ~=A*. by universal coefficients. The dual algebra A* is a Hopf algebra with antipode c coming from inversion in the group G, and A is a right Hopf module over A*: the module structure is given by A A* ! A, the adjoint map of the coproduct _ : A ! A A, and the comodule structure by A ! A A*, the adjoint map of the product on A. Let P (A) denote the Fp-vector space of primitives of A as an A*-comodule, i.e. n fi o P (A) = a 2 A fifiax = affl(x) for allx 2,A where ffl is the augmentation H*(G+ ) ! H*(S0). Then (cf [Par71 ]), we have a splitting ~= * A ! P (A) A 16 as right A*-Hopf modules, given by ____//_ * _id__//_ * * id_c_id// * * __~_id_// * A [ [ [A[ [A[ A A A A A A A AOO (4.1) [ [ [ [ | [ [ [ [ [ [ | [ [ [[ [ [ | [ [ [ [ [ [--P (A) A* Since A is finite dimensional, it follows that dim P (A) = 1. The assertion of Lemma 19 follows. 2 We will show later (Proposition 26) that for G a p-compact group, this map is realizable as a map of spectra. An algebra like H*(G; Fp), which, as a module, is isomorphic to a suspension of its dual, is called a Frobenius algebra. Lemma 20 (Moore-Peterson [MP73 ]) If A is a finite-dimensional Frobe- nius algebra over a field, then the class of its projective modules coincides w* *ith the class of its injective modules. 2 Proposition 21 SG is homotopy equivalent to a HZ=p-local sphere of dimen- sion d. PROOF. It is enough to know that SG has the mod p homology of a sphere because the proof of Lemma 7 produces a p-equivalence Sd ! SG in that case. To see that SG has the correct homology, we will use a spectral sequence associated to the cosimplicial spectrum op G 1 ( 1 G+ )hoG = map (EG+ , G+ ), where EG = map (-, G) = 1 G is the usual simplicial space with Gn+1 in dimension n. The E2-term is given by E2p,q= Hp(map G(G(q+1), 1 G+ ); Fp), and by the Lemma below, this spectral sequence collapses at the E2-term with 8 < 0; p 6= 0 or q 6= d Ep,q2= : Fp; otherwise. 17 and converges strongly. Therefore and H*(SG ) = H*(Sd). 2 This proves the first part of Theorem 2. Lemma 22 Let k 2 N0[ {1}, and let EG(k)+be the G-equivariant k-skeleton of the simplicial space EG+ . Then 8 < Fp; n = d < k Hn(map G(EG(k)+, 1 G+ )) ~=: 0; n < k and n 6= d No statement is made about the homology groups beyond degree k. By G- equivariant k-skeleton we mean the truncation of the simplicial space EG+ at the kth stage. From now on, until the end of this section, all homology groups are taken with coefficients in Fp. PROOF. Since EGn = Gn+1, we have that (map G(EG+ , 1 G+ ))n = map G (Gn+1+, 1 G+ ). The evaluation map map G(Gn+1+, 1 G+ ) ^ Gn+1+! 1 G+ induces a natural map Hn(map G(Gn+1+, 1 G+ ) ! Hom nH*(G+)(H*(G+ ) (n+1), H*( 1 G+ )), where Hom n stands for module homomorphisms that raise degree by n. This map is an isomorphism because the following diagram commutes: Hn(map G(Gn+1+, 1 G+ ))_____//HomnH*(G+)(H*(G+ ) (n+1), H*( 1 G+ )) |~| |~| |fflffl fflffl| Hn(map (Gn+, 1 G+ )) Hom n(H*(G+ ) n , H*( 1 G+ )) |~| |~| |fflffl fflffl| Hn((DG+ )^n ^ 1 G+ ) ______~___//(H*(DG+ ) n H*( 1 G+ )) n. The coboundary operators are induced by the simplicial operators on H*(G+ ) o from the bar construction on H*(G+ ). Hence Hn(map G(EG(k)+, 1 G+ )) is the 18 group of homomorphisms from a truncated projective resolution of k over H*(G) to H*(G). Associated to the tower map G(EG(k)+, 1 G+ ) ! . .!.map G (EG(j)+, 1 G+ ) ! ! . .!.map G (EG(0)+, 1 G+ ) = 1 G+ , we therefore obtain a spectral sequence with E2-term 8 < Extp,q (Fp, H*(G+ )); q < k (k) E2p,q= : H*(G+) =) Hp+q(map G(EG+ , 1 G+ ). 0; q > k Because of Lemmas 19 and 20, H*(G+ ) is injective as a module over itself, and hence E2p,q= 0 for q > 0 and q 6= k. On the other hand, E2p,0= E1p,0= Hom qH*(G+)(Fp, H*(G+ )) = P (H*(G+ )) with the notation of Lemma 19, and hence 8 < Fp; n = d Hn(map G(EG(k)+, 1 G+ )) ~=: 0; n < k and n 6= d 2 5 Self-duality for p-compact groups 5.1 Two Lemmas on restricted homotopy fixed points Lemma 23 For a sub-p-compact group H < G, there is a coarse G-equivalence op G ^H SH - ~! ( 1 G+ )hoH . PROOF. First note that as (G x Hop)-spectra, 1 G+ ' G ^H 1 H+ , where on the right hand side, H acts on the right factor from the right and G acts on the left factor from the left. 19 We therefore have a map op 1 G ^H SH = G ^H map H (EH+ , H+ ) op 1 1 hoHop - ! map H (EH+ , G ^H H+ ) = ( G+ ) . This map is clearly G-equivariant, and it is a weak equivalence because G and H are nonequivariantly dualizable. 2 If X 2 (G x Gop)S and Y 2 HopS, we have G-equivariant homotopy equiva- lences (given by shearing maps) G ^H X ' G=H+ ^ X and op map H (Y ^ G+ , X) ' map (Y ^H G, X) In particular, if Y 2 (G x Gop)S, we have op map H (Y ^ G+ , X) ' map (Y ^ G=H+ , X). (5.1) Lemma 24 Let H < G be as above. Then there is a coarse G-equivalence op ~ (DG+ ) hoH -! D(G=H+ ), natural on subgroups of G. PROOF. The map is the following composite of coarse G-equivalences, all of which are natural: op H (DG+ )hoH = map (EH+ , DG+ ) ' map H(EG+ , DG+ ) -! map H(EG+ ^ G+ , S0) f 0 -! map (EG+ ^ G=H+ , S ) g -! D(G=H+ ). 20 For the first homotopy equivalence, we use that EG is a valid model for EH. f is a G-equivariant homotopy equivalence by (5.1). Since EG+ has the usual right action and a trivial left action, the map S0 ! EG+ is a left G-homotopy equivalence, and hence so is g. 2 5.2 Absolute Poincar'e duality Denote by Gc the suspension spectrum of G with G acting by conjugation. For G a Lie group, SG can be identified with the one-point compactification of a neighborhood of the identity in G; this identification is G-equivariant if we equip G with the conjugation action. The following lemma shows that such a öl garithmä lso exists for p-compact groups, at least up to a coarse G-equivalence. Lemma 25 For every p-compact group G, there is a G-spectrum E(G), a natural coarse G-equivalence E(G) ! (Gc)+ , and a G-equivariant retraction E(G) ! SG . Remark: An equivariant retraction X ! Y means two equivariant maps Y ! X ! Y such that the composite is a coarse G-equivalence. PROOF. The auxiliary spectrum E(G) is defined as op 1 E(G) = map G (EG+ , G+ ^ DG+ ). Consider 1 G+ as a (G x Gop)-spectrum by left and right multiplication. Then the diagonal map 1 G+ ! 1 G+ ^ 1 G+ is (G x Gop)-equivariant and has an equivariant adjoint 1 G+ ^ DG+ ! 1 G+ . (5.2) Similarly, the (G x Gop)-equivariant projection map to the first factor 1 G+ ^ 1 G+ ! 1 G+ 21 has an equivariant adjoint 1 G+ ! DG+ ^ 1 G+ . The composite 1 G+ ! 1 G+ ^ DG+ ! 1 G+ (5.3) is a weak equivalence. Taking homotopy fixed points with respect to Gop = 1 x Gop G x Gop on the left hand side of (5.2) yields op 1 ~ Gop 1 E(G) = map G (EG+ , G+ ^ DG+ ) - ! map (EG+ ^ G+ , G+ ) (2.2) - ~! map (EG 1 + , G+ ) - ~! map (S0, 1 G 1 + ) ' G+ (5.4) As in Lemma 24, the map induced by S0 ! EG+ is indeed a G-homotopy equivalence because the left G-action on EG+ is trivial. In fact, all maps but the first one are G-homotopy equivalences. We have to check that the G-action on E(G) corresponds to the conjugate action on 1 G+ . op 1 The action of G on M = map G (EG+ ^ G+ , G+ ) is given by (g.f)(x ^ fl) = gf(x ^ g-1 fl) (g 2 G, f 2 M, x 2 EG+ , fl 2 G+ ). The induced action of G on map (EG+ , 1 G+ ) is (g.f)(x) = gf(xg)g-1 (g 2 G, f 2 map (EG+ , 1 G+ ), x 2 EG+ ) since op 1 1 map G (EG+ ^ G+ , G+ ) ! map (EG+ , G+ ) f 7! x 7! f(x, 1) g.f 7! x 7! gf(x, g-1 ) = gf(xg, 1)g-1 . The restricted G-action on map (S0, 1 G+ ) becomes (g.f)(x) = gf(x)g-1 since S0 has the trivial G-action, and the G-action on 1 G+ is indeed by conjugation. 22 op Applying (-)hoG to (5.3)yields the desired retraction E(G) ! SG . 2 Proposition 26 Regard the G-spectrum SG as a (G x Gop)-spectrum with trivial Gop-action. Then there is a coarse G x Gop-equivalences SG ^ DG+ -'! 1 G+ On Gop-homotopy fixed points, these maps make the following diagram com- mute: op ' hoGop ' hoGop SG = ( 1 G+ )hoG Zoo___(SG ^ DG+ ) oo___SG ^ (DG+ ) ZZZZZZZZZZZZZZZ ZZZZZZZZZZZZZZZZZZ ' Lemma|24 ZZZZZZZZZZZZZZZZZZ | ZZZZZZZZZZZZZZZZfflffl| SG ^ S0 PROOF. We will have to deal with spectra with three G-actions, and for ease of notation, for a (G x Gop)-spectrum X, I will denote by aXbcthe spectrum X with the left action a and the two right actions b and c, where a, b, c, are one of the following: o `O' denotes a trivial action o `l' denotes the action from the left _ if this symbol appears on the right then G acts by inverses from the left o `r' denotes the action from the right _ if this symbol appears on the left then G acts by inverses from the right The main ingredient is a shearing map l( 1 G+ )O l r sh l 1 O O l r ^ (DG+ )O -! ( G+ )r ^ (DG+ )r. (5.5) which is adjoint to l( 1 G+ )O O 1 l l 1 O l 1 r r ^ ( G+ )r- ! ( G+ )r ^ ( G+ )O g ^ h 7! g ^ hg. This map is clearly a weak equivalence, and it is straightforward to check that it is (G x Gop x Gop) -equivariant as claimed. 23 By passing to homotopy orbits with respect to the O Ooaction of Gop in (5.5), we obtain a (G x Gop)-equivariant homotopy equivalence i j hoGop i jhoGop l( 1 G+ )O l r _____//l 1 O O l 0 r ^ (DG+O)OO ( G+ )r ^ (DG+ )r =: E (G) ' || ' || | fflffl| l(SG )O ^ l(DG+ )r l( 1 G+ )r The underlying spectrum of E0(G) is the spectrum E(G) of Lemma 25. It is easy to see that with the remaining operations, the map E0(G) ! 1 G+ , described in (5.4), is (G x Gop)-equivariant. For the assertion about Gop-homotopy orbits, observe that by changing the order of taking Gop-homotopy orbits, we have a large commutative diagram ___________________________________________* *_____________________________________________________________________@ ___________________________________________________* *_____________________________________________________________________@ ________________________________________________________* *_____________________________________________________________________@ lSG = lGhoGop oo_____'_____lSG ^ O(DG)hoGop ___'__________________* *_______//_lSG OO | | || ' | ' | || i | j O o i fflffl|j O |||| lGO O l ho ooo_'__ lO O hoGop ho o _'__//_l r ^ (DG)rOO Gr ^ (DG)OOO SG || ' sh|| ' sh|| |||| i | j hoO o i | j hoO || lGO l r ooo_'___ l l hoGop o__'__//l r ^ (DG)OOO Gr ^ (DG)OOO SG | ' || ' || i | jhoO o | op l(SG )O ^ l(DG)r oo_'____lSG_^ l(DG)hoG For space reasons, the disjoint basepoint for 1 G and DG have been omitted as well as the suspension functor 1 for G. The important, if trivial, observation is that the shear map becomes homotopic to the identity when passing to O oO-homotopy orbits on the DG+ factor. The diagram claimed to be commutative in the proposition is the öb undaryö f the diagram above. 2 24 5.3 Relative Poincar'e duality Corollary 27 For any sub-p-compact group H of G, there is a zigzag of coarse G-equivalences G ^H SH oo'_//_____________D(G=H+ ) ^ SG This zigzag is natural in the following sense: for any chain of p-compact groups K < H < G, the following diagram commutes: op ' ' ( 1 G+ )hoH oo___G ^H SH oo__//____________D(G=H+ ) ^ SG |res| |D(proj)^id| fflffl|op fflffl| ( 1 G+ )hoK oo'__G ^K SK oo'_//_____________D(G=K+ ) ^ SG PROOF. From Proposition 26, we have a coarse (G x Hop)-equivalence DG+ ^ SG ! 1 G+ . Applying Hop-homotopy orbits turns coarse (GxHop)-equivalences into coarse G-equivalences, and since the right actions on SG and SH are trivial, we obtain op ' hoHop ' (DG+ ^ SG ) hoH oo___DG+ ^ SG ____//_D(G=H+ ) ^ SG |'| fflffl|op ( 1 G+ )hoH oLem'mao23G_^H SH For naturality, consider the following diagram: op hoHop 1 hoHop D(G=H+ ) ^ SG oo___DGhoH+ ^ SG _____//(DG+ ^ SG ) ____//_( G+ ) | | | | | | | | fflffl| opfflffl| fflffl|op fflffl|op D(G=K+ ) ^ SG oo___DGhoK+ ^ SG _____//(DG+ ^ SG )hoK ____//_( 1 G+ )hoK The left hand square commutes by Lemma 24, the other two for trivial reasons. 2 5.4 Definition of the transfer Proof of Theorem 2: SG was constructed in Section 4. We obtain a (nonequiv- ariant) map _ 1 hoGop 1 hoHop' t: SG = ( G+ ) _____//( G+ ) __f_//_G ^H SH 25 coming from restricting from G- to H-homotopy fixed points. Here, f is the nonequivariant homotopy inverse of the coarse G-equivalence given by Lemma 23. By Lemma 6, there is also a G-equivariant map ~t: EG+ ^ SG - ! G ^H SH such that the composite SG ! EG+ ^ SG ! G ^H SH is homotopic to ~t, and ~tis unique up to homotopy with this property. To finish the proof of Theorem 2, we need to show that ~t*: Hd(SG ; Fp) ! Hd(G ^H SH ; Fp) is an isomorphism for d = dim G. This now follows easily from Corollary 27: The map ~tis by construction the composite SG ! D(G=H+ ) ^ SG !~ G ^H SH . Since the first map is an isomorphism in Hd(-; Fp), so is the composite. 2 The first part of Theorem 3 claims that H*(BGg; Fp) is a Thom module over H*(BG+ ; Fp). This follows from the spectral sequence E2 = H*(BG+ ; H*(SG ; Zp)) =) H*(EG+ ^G SG ; Fp) = H*(BGg; Zp). 2 Definition 28 For a monomorphism H < G of p-compact groups, the trans- fer map tG,H is given by applying G-homotopy orbits to the G-equivariant map ~t. The domain of ~tis EG+ ^G (EG+ ^ SG ), which by Lemma 6 is homotopy equivalent to BGg. For the functoriality, it is sufficient to notice that the following diagram of G-equivariant maps commutes: 26 SG _____//MMM( 1 G+ )hoHopoo____G ^H SH MMM | MMM | M&&M op fflffl|op ( 1 G+ )hoKiiTT G ^H ( 1 H+O)hoKO TTTTTT | TTTTT | TT | G ^K SK That is a less than remarkable statement since no two maps are composable. But all of the maps going left or up or both are coarse G-equivalences, and the diagram stays (non-equivariantly) homotopy commutative if we invert them. By its definition, the commutativity of SG _____//_G ^H SG | | | | fflffl| fflffl| BGg ______//_BHh as claimed in Theorem 3 is immediate. The next section will be devoted to identifying the transfer map on the cate- gory of HZ=p-localizations of compact Lie groups and monomorphisms. 6 Identification of the transfer map Using the construction of t for {1} < G, we have a commutative diagram coming from the natural transformation id ! (-)hoG : SG ______// 1 G+ (6.1) | | | | fflffl|t fflffl| BGg _____//B{1}+______S0 We will now identify t with the transfer map from the introduction in the case where G is of Lie type. First note that in (6.1), the composite map Sd ! S0 is indeed the same as the Thom-Pontryagin construction on G if G is a Lie group: The Thom-Pontryagin construction on G is given by the composition of maps S0 ! DG+ ' S-d ^ G+ ! S-d, 27 where the first map is a desuspension of the map from the embedding sphere to the Thom space of the normal bundle of G, which is DG; by Proposition 26 or since the tangent bundle of G is trivial, this is equivalent to a desuspension * *of 1 G+ ; and the second map is the projection of G+ to S0, the map classifying the (trivial) stable normal bundle of G. Using the G-equivariant isomorphism from Proposition 26, we have (-)hoG SG _______//KKO1OG+_____//S0 KKK _____ KKK _~___ KK%%fflffl___ SG ^ DG+ The bottom composition is the Thom-Pontryagin construction, the upper one the map from (6.1). 6.1 An alternative construction of the transfer map To show that t agrees with the Umkehr map not only on the bottom cell, we will compare its definition to another, equally general construction, reminis- cent of Dwyer's construction of the Becker-Gottlieb transfer in [Dwy96 ]. This will be equivalent to the classical construction of the Umkehr map in the Lie case. Let H < G be p-compact groups. The quotient 1 G=H+ is dualizable, and the projection D(G=H+ ) ! S0 onto the top cell is equivariant and has a section ff; however, ff is not G-equivariant unless H = G. But we do get an equivariant map if we "free up" the G-action on S0: consider the following diagram of G-equivariant maps: 1 EG+ ~___//_S0___//map( 1 G=H+O,O 1 G=H+ ) '''|| | D(G=H)+ ^ 1 G=H+ 1^i___//_D(G=H)+ ^ S0. The map ß is the projection 1 G=H+ = 1 G=H _S0 ! S0, and j is a coarse G-equivalence. Therefore, by Lemma 6, there exists an equivariant lifting 1 EG+ ! D(G=H+ ) (6.2) which is nonequivariantly homotopic to the map 1 EG+ ! S0 ff!D(G=H+ ). 28 In the case where H and G are (localizations of) Lie groups, the homotopy orbit space of D(G=H+ ) under this G-action is the (localization of the) Thom space of , the normal bundle along the fibers of BH ! BG. This follows from the observation that BH = EG xG G=H, and that stably, the normal bundle along the fiber is the fiberwise Spanier-Whitehead dual, i.e. = EGxG D(G=H). Hence its Thom spectrum is EG+ ^G D(G=H+ ), as claimed. By passage to G-homotopy orbits in (6.2), we therefore obtain a map 1 BG+ ' EG+ ^G 1 EG+ ! BH-g=h, where BH-g=h denotes the Thom spectrum of the virtual inverse of the adjoint bundle of G, pulled back to BH, modulo the adjoint bundle of H. This is the Lie theoretic model of the normal bundle along the fibers. Returning to the case of a general p-compact group, we now introduce a "twist- ing" by smashing source and target of the map with SG : EG+ ^ SG _____//map( 1 G=H+ , S0) ^ SG __'__//D(G=H+ ) ^ SGCoo'or./27/_______* *______G ^H SH By Lemma 6 and since EG+ ^ SG is a free G-spectrum, we obtain a G-map (unique up to homotopy) ~t0: EG+ ^ SG ! G ^H SH , and passing to G-homotopy orbits, we obtain: t0: BGg = EG+ ^G SG ! EH+ ^H SH = BHh. Lemma 29 Let H < G be p-compact groups. Then ~t' ~t0: EG+ ^ SG - ! G ^H SH PROOF. We have to show that the following G-equivariant diagram com- mutes: EG+ ^ SG __(6.2)//_D(G=H+O)O^_SG ____ ~t|| ~ ______ fflffl|~ fflffopl___ G ^H SH oo___(DG+/^/SG_)hoH___________ Since EG+ ^ SG is a free G-spectrum, there is by Lemma 6 a G-equivariant map going diagonally op EG+ ^ SG ! (DG+ ^ SG )hoH 29 and making the upper right triangle of the diagram commute. The commuta- tivity of the lower left triangle then follows from the observation that in the commutative diagram ~ op hoHop SG oo___(DG+ ^ SG )hoG ____//_(DG+ ^ SG ) , | | | | | | fflffl| fflffl|op fflffl|op SG ________( 1 G+ )hoG ________//_( 1 G+ )hoH the left hand map is the identity by Proposition 26. 2 Conclusion of the proof of Theorem 3 The previous lemma implies the third part of the theorem (namely, that t O Lp ' Lp O (-)! on compact Lie groups and monomorphisms). Indeed, by applying G-homotopy orbits to the diagram in Lemma 29, the map induced by ~t0is homotopic to t, and the preceding discussion shows that the former map is the classical Umkehr map in the Lie case. 2 7 Computational methods In this section, I will describe a general method for computing the homotopy class represented by a p-compact group by constructing a representing cycle in the Adams spectral sequence for a complex oriented cohomology theory E. Let G be a simply connected d-dimensional p-compact group of rank r with maximal torus T . We want to identify the maps the following diagram induces in the E2-term for the E-cohomology ASS: Sd ! BGg ! BT t! S0 7.1 The S1-transfer The right-hand map is a suspension of the r-fold smash product of the S1- transfer map ø : CP+1 ! S-1. 30 It is well-known that the homotopy fiber of this transfer map is the spectrum CP-11, the Thom spectrum of the inverse of the universal line bundle on CP 1, the fiber inclusion CP-11! CP+1 being the obvious projection map onto the 0-coskeleton. For a complex oriented cohomology theory E and a finite spectrum X, there is an Adams-Novikov spectral sequence E2 = Ext(E*(X)) =) [X, LE S], where, for an (E*, E*E)-comodule A, Ext(A) is a shorthand for ExtE*E (E*, A). We will now first restrict to the case of finite dimensional projective spaces and study the map CP+m ! S-1 as a map of E2-terms of this ANSS Ext(E*(S-1)) _______+3_ß*(LE S1) | | | | fflffl| fflffl| Ext(E*(CP+m)) _____+3_[CP+m, LE S]. By a change of rings isomorphism, this spectral sequence is isomorphic to the one associated to the Hopf algebroid (Am , m ) = (E*(CPnm), (E ^ E)*(CPnm)) . Note that Am represents the following functor: 8 9 >>> E* !ffR, >> >>> fi >>> < fifif is a function modulo degree m + n + 1 on the formal >= R 7! > (ff, f) fifigroup law on R given by the image of the universal formal >>> fi * * >>> >>: group law under MU ! E ! R such that f vanishes to >>>; the nth order at the identity. Similarly, m represents isomorphisms of such data. Hence, for E = MU, (Am , m ) classifies formal groups with an m + n + 1-truncated function on it that vanishes to the nth order at the identity. This interpretation makes it easy to compute the structure maps of (Am , m ). First assume that n = 0. Pick coordinates z such that E*(CP0m) = E*[[z]]=(zm+1 ) and (E ^ E)*(CP01) = (E ^ E)*[[z]]=(zm+1 ). Since there is a map of Hopf algebroids (E*, E*E) ! (Am , m ), we only need to determine jL(z) and jR (z). We can make jL(z) = z by choice of coordinates; 31 then, jR (z) will be the image of the universal isomorphism m-1X bizi+1 2 (MU ^ MU)*(CP0m) i=0 in m . As usual, MU* = Z[mi] and MU*MU = MU*[bi]. If n 6= 0, E*(CPnm) is still a free E*-module, generated by {zn, zn+1 , . .,.zn* *+m }, and the above formula for jR is correct when interpreted as jR (zi) = jR (z)i. For our purposes, it would be easier to use BP -theory instead of MU since we are working in a p-local setting anyway. However, it only affects the complexity of the computations, not the method. Assembling all spectral sequences for varying m 0, we obtain towers .. . . . .. | | | | fflffl| fflffl| Ext m+1(Am+1 , Am+1 )_____+3_[CPnm+1, LE S] | | | | fflffl| fflffl| Ext m (Am , Am )________+3_[CPnm, LE S] | | | | fflffl| fflffl| .. . . .. The inverse limit of the left tower is not quite the Ext term associated to the Hopf algebroid (A, ) = (E*(CP 1), (E ^ E)*(CP 1)). This is due to the fact that (E ^ E ^ E)*(CP 1) fi A (the left hand side is a completion of the right hand side). Similarly, the inverse limit of the tower on the right hand side is not quite [CPn1, LE S]. It does not include the phantom maps. Coming back to the problem of determining the induced map of the S1-transfer on E2-terms, we look at the cobar construction functor Bn(M) = M E* (E*E) E*n for an (E*, E*E)-comodule M. 32 0OO 0OO 0OO | | | | | | | | | Ext( -1S-1)OOoo_______Ext(CP-11)Ooo_______ExtO(CP01)OO | | | | | | | | | Z*(E* -1S-1)OO oo_____Z*(E*CP-11)OOoo_____Z*(E*CP01)OO oo____0 d1|| d1|| d1|| | | | 0 oo___B*-1(E* -1S-1)OO oo___B*-1(E*CP-11)OOoo___B*-1(E*CP01)OO oo___0 | | | | | | | | | Z*-1(E* -1S-1)OO oo___Z*-1(E*CP-11)OOoo___Z*-1(E*CP01)OO oo___0 | | | | | | | | | 0 0 0 Fig. 7.1. This diagram admits a snake map. Since E*, E*E, and H*(CP 1) are concentrated in even dimensions, and since B is an exact functor on flat E*-modules, we have a short exact sequence of B(E*, E*E)-modules 0 B(E* -1S-1)) B(E*(CP-11)) B(E*(CP01)) 0. If we denote by Z*(X) B*(X) the cycles under the cosimplicial differential d1, we have a diagram as shown in Figure 7.1. It follows from the snake lemma that the kernel of the top right map is the image under the snake map Z*-1(E*S-2) ! Ext(CP01), which is, by following through the diagram, the image of d1|E*{z-1}. Now if T is a p-compact torus, the transfer map in cohomotopy is simply the r-fold smash product of the map represented at the E2-level by d|E*{z-1}. 7.2 The map Sd ! BGg ! rBT We will first study the effect of this map on rational cohomology. By Theorem 12, H*Qp(BG) = H*Qp(BT )W(G) is always a polynomial algebra. Proposition 30 For a p-compact groups G with maximal torus T , the fol- 33 lowing diagram commutes: H*Qp(BT+ )__proj_//H*Qp(BT+ ) = (H*Qp(BG))Cor._13_H*Qp(G=T+ ) |-rt*| || fflffl| fflffl| H*Qp( -rBGg) _____~=___//H*Qp(BG+ ){ø }__fi7!'__//H*Qp(Sd-r), where ø is the Thom class of BGg, and ' is the generator in H*(Sd-r). This allows us to compute the effect of the map Sd ! BGg ! rBT fiber in cohomology (this is the bottom composition of maps in the diagram) by simply evaluating at the image in H*Qp(BT+ ) = (H*Qp(BG)) of the fundamental class of G=T . Proof of the proposition By the construction of the transfer map, we have a commutative diagram G ^TOSrO_____// rBT+OO, | | | | | | Sd _________//BGg and by Theorem 2(2), the left hand map Sd ! rG=T+ is an isomorphism in the top homology group. Desuspending r times and applying H*Qpyields the commutativity of the diagram of the proposition. 2 Now let E be a HZ=p-local complex oriented torsion free cohomology theory. Denote by EQp the cohomology theory X 7! E*(X) Zp Qp. We have E*(CP 1) = E*[[z]] ,! E*Qp(CP 1), and the same is true for E ^ E. Hence for computing a cobar representative, we can work with rational coefficients and always hope that in the end of our computations, everythings turns out integral. To compute this, we can use the Chern characters exp : E*Qp(X) ! H*Qp(X) ^E* and exp : (E ^ E)*Qp(X) ! H*Qp(X) ^ß*(E ^ E) 34 which, for X = CP+1, is the exponential map for the formal group law associ- ated with E and an isomorphism, and for X = BT+ , a tensor power thereof. The smash product is formed in the HZ=p-local category, as always. This induces an isomorphism of (EQp *, (E ^ E)Qp *)-modules B(exp ) : B(E*QpBT+ ) ! B(E*) ^H*Qp(BT+ ). We have a commutative diagram B(E*QpBT+ ) oo___________B(E*BT+o)o m PPPP mmmm | | PPPP mmmm | | PPP vvmmmm || || PP(( B(E*Qp) oo_____________|_______________________|______________B(E*)oo hhQQQ | | nnn66 QQQQ | | nnn QQQQQ | | nnnn Q fflffl| fflffl|nnn B(E*Qp) H*(BT+ ) oo___B(E*)oo H*(BT+ ) So, to evaluate the class in B(E*(BT )) computed in the first part, we apply B(exp ) to it and obtain a class in B(E*) H*Qp(BT ), which we then eval- uate at the image of the fundamental class [G=H] in (HQp )d-r(G=H+ ) ! (HQp )d-r(BT+ ). This class then must actually be integral. 8 The family no. 3 of groups ~m The p-compact group ~m , for p ~=1 (m), has rank 1 and Weyl group W Zxp a cyclic subgroup of order m of the p-adic units. It is a nonmodular group and can therefore be constructed as B~m = Lp (K(Zp, 2) xW EW ) . Therefore, H*(B~m ; Zp) = Zp[[z]]W , where a W acts on z by multiplication. This shows that H*(B~m ; Zp) = Zp[[zm ]] ,! Zp[[z]]. The fundamental class [~m =T ] 2 H*Qp(BT )=(H*Qp(B~m )) = Qp[z]=(zm ) is zm-1 , and we conclude that ~m has dimension 1 + 2(m - 1). It is straight- forward to see that for m < p - 1, ~m cannot represent a nontrivial ho- motopy class in the p-stems because (ßsn)(p)= 0 for 0 < n < 2p - 3. But (ßs2p-3)(p)= Z=p{ff1}, and we will see that ~p-1 represents this class. 35 In the p-completed BP -spectral sequence for ß*(CP-11), jR (z) = z + t1zp + O(zp+1), hence jR (z-1 ) = z-1 - t1zp-2 + O(zp-1). Applying the Chern character to this power series does not change it up to O(zp-1), and hence [~p-1] is the coefficient of zp-2 of this series, which is t* *1. Lying in filtration 1, t1 represents the homotopy class ff1. 9 Some exceptional cases 9.1 The 5-compact group no. 8 The pseudo-reflection group G which is no. 8 in Shephard and Todd's list has order 96 and is generated, as a complex reflection group, by the two reflections 0 1 0 1 1_ i_ 1_ i_ B@-i 0CA and B@2- 2 -2 + 2 CA. 0 1 1_2- i_21_2- i_2 The ring of invariants Z5[x1, x2]G is polynomial because 5 does not divide the order of G; a straightforward calculation shows that it is generated by the polynomials ~ = x81+ 14x41x42+ x82 and = x121- 33x81x42- 33x41x82+ x122. Hence H*(BG; Z5) = Z5[~, ], and by Proposition 15 the cohomology of G=T is given by H*(G=T+ ; Z5) = H*(BT+ ; Z5) = (H*(BG; Z5)). A Gröbner basis calculation shows that the top class in H36(G=T ; Z5) is 1 3 15 1 15 3 x71x112= -x111x72= -___x1x2 = ___x1 x2. (9.1) 13 13 We will use the 5-primary BP -spectral sequence for ß*(CP-11) to determine the homotopy class G represents. In this spectral sequence, 36 i j i j jR (z) = z - t1z5 + 5 t12+ t1v1 z9 + -35 t13- 12 t12v1- t1v12 z13 i j + 285 t14+ 137 t13v1+ 21 t12v12+ t1v13 z17 +O(z21) and hence i j i j i j d1(z-1 ) = - t1z3 + 4 t12+ t1v1 z7 + -26 t13- 10 t12v1- t1v12 z11 i j + 204 t14+ 106 t13v1+ 18 t12v12+ t1v13 z15 +O(z19). Applying the Chern character to this class yields ` 8 t v ' _ 78 t2 v 78 t v 2! f(z) = -t1z3 + 4 t12+ ___1_1_ z7 + -26 t13- ____1__1_- ____1_1_ z11 5 5 25 _ ! 816 t13v1 1224 t12v12 816 t1v13 15 + 204 t14+ _________ + ___________ + _________ z 5 25 125 +O(z19). We need to evaluate the class f(z) f(z) at the classes given in (9.1)and add them up. This yields: [G]= -204t1 t41- 808t21 t31- 1208t31 t21- 604t41 t1 -160v1t1 t31- 480v1t21 t21- 320v1t31 t1 -48v21t1 t21- 48v21t21 t1. By adding a suitable boundary, namely d1(4t51+ 45v1t41+ 34v21t31+ 10v31t21+ v41t1), we see that this class is homologous to t1 t41+ t41 t1 + 2(t21 t31+ t31 t21), which is the representative of fi1 in the ANSS. 37 9.2 The 3-compact group Za2 (no. 12) The Weyl group W of the modular group Za2 constructed by Zabrodsky [Zab84 ] is generated by the two matrices 0 1 0 1 _1_p_i_p_ 1_ i_ 1_ i_ B@- 2 2CA and B@ 2 + 2 2 + 2CA. -_i_p_21_p_2 -1_2+ i_21_2- i_2 Although 3 | #W , the ring of invariants Z3[x1, x2]W is polynomial, generated by the polynomials ~ = x81+ 14x41x42+ x82 and = x51x2 - x1x52. We find that the top class in H36(G=T+ ; Z3) is 1 12 1 12 x41x82= x81x42= -___x1 = ___x2 . (9.2) 15 15 In the 3-primary BP -ANSS, logarithm, exponential, and universal isomor- phism are all odd power series; hence, evaluation at the class above yields 0 without further computations. 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