Some Background Material

Some background material Steve Mitchell September 24, 2011 This is background material for my topics course \Equivariant cohomology of finite group actions". Much of it will be discussed also in the lectures. 1 Topological interpretation of group cohomology Let G be a discrete group, M a G-module. Then group cohomology with coefficients in M is by definition H∗(G; M) = Ext∗ ( ;M): ZG Z Here the Ext groups can be computed using either an injective resolution of M, or a projective resolution of the trivial module Z. Using the former version, we see that the group cohomology is given by the right derived functors of the invariants functor M 7! M G. On the other hand, there is the following topological interpretation in terms of the classifying space BG: Theorem 1.1 Let G be a discrete group. Then H∗(G; M) ∼= H∗(BG; M), where M is the local coefficient system associated to M. We remark that there are various definitions of cohomology with local coefficients, but the simplest goes like this: Let X be a path-connected space with a universal cover X~. Then a local coefficient system is just a module M over the fundamental group π1X, and we define ∗ ∗ ~ H (X; M) = H (HomZπ1X (C∗X;M)): Here C∗ denotes the singular chain complex. With this definition the theorem is almost a tautology, since for discrete groups the universal cover of BG is the contractible space EG, whose singular chain complex therefore gives a free resolution of Z: If M is a trivial G-module (meaning the action of G on M is trivial), we just get ordinary cohomology with coefficients in M. This is the case we are primarily interested in, with M = Fp. This topological interpretation is very powerful. Among other things, it allows us to bring compact Lie groups and their classifying spaces into the picture. The advantage of this is that for classical groups such as G = U(n) or G = O(n), the cohomology of BG has a much simpler structure than that of a typical finite group, and there are functo- rial generators: Chern classes, Stiefel-Whitney classes, Pontrjagin classes. In particular 1 we can define Chern/Stiefel-Whitney/Pontrjagin classes of a representation as follows: If θ : G−!U(n) is a complex representation of G, ck(θ) = ck(ξ), where ξ is the vector bundle defined by Bθ : BG−!U(n). Explicitly, ξ can be realized as the Borel construction n EG ×G C . Stiefel-Whitney/Pontrjagin classes are defined similarly, using real representa- tions. This construction is especially useful for computing induced maps. Suppose we have a homomorphism φ : G−!H and want to compute φ∗ : H∗BH−!H∗BG. If α 2 H∗BG ∗ has the form α = ck(θ)) for some representation θ, then φ α = ck(θ ◦ φ): We also recall one crucial distinction that must be kept in mind: Let EG denote the universal principal G-bundle over BG as usual, so EG is contractible. If G is discrete, then EG is the universal cover and BG is also known as a K(G; 1)-space, i.e. π1BG = G and πnBG = 0 for n > 1. But if G is not discrete, say G is a compact Lie group of positive dimension, then EG is certainly not the universal cover and BG is not a K(G; 1). Furthermore, we no longer have the purely algebraic description of H∗BG as an Ext group. Some basic facts about BG (G a compact Lie group, although all of these results hold much more generally): Proposition 1.2 Suppose θ : G−!G is an inner automorphism. Then Bθ : BG−!BG is homotopic to the identity. In particular, Bθ∗ is the identity on H∗. Corollary 1.3 Let H be a closed subgroup of G. Then the conjugation action of NGH on ∗ ∗ ∗ H BH factors through W = NGH=H, and the image of H BG−!H BH lies in the ring of invariants (H ∗ BH)W . Proposition 1.4 If H is a closed subgroup of G, then there is a fiber sequence G=H−!BH−!BG; where BH−!BG is induced by the inclusion H ⊂ G. If H is normal in G, inclusion H ⊂ G and projection G−!G=H induce a fiber sequence BH−!BG−!B(G=H): Here A−!B−!C is a fiber sequence if there is a homotopy-commutative diagram ABC- - ? ? ? - - XYZq such that the vertical maps are weak equivalences and q is a Serre fibration with fiber X, or an analogous diagram with the vertical arrows reversed. Thus we can treat A−!B−!C as though B−!C was a fibration with fiber A; in particular we have a Serre spectral sequence H∗(C; H∗A) ) H∗B. In the first case of the proposition, the spectral sequence for G finite is uninteresting; by Shapiro's lemma it degenerates to the identity H∗H = H∗H. But it will be very useful in the case G is a connected Lie group such as U(n). When G is finite, the spectral sequence associated to the second fiber sequence of the proposition is the same as 2 the Hochschild-Serre spectral sequence of the group extension H−!G−!G=H. The latter can be constructed by pure homological algebra, most elegantly by using Grothendieck's composite functor spectral sequence (the G-invariants functor M−!M G factors as M 7! M H 7! (M H )G=H ). 2 The transfer Throughout this section G is a finite group, H is a subgroup of G, and M is a G-module. We have a restriction homomorphism on group cohomology i(H; G)∗ : H∗(G; M)−!H∗(H; M): In topological terms, i∗ is induced by the map on classifying spaces BH−!BG. The transfer is a homomorphism going the other way: τ(H; G): H∗(H; M)−!H∗(G; M): Thinking of group cohomology as derived functors of the invariants functor, the transfer is H G P induced by the obvious abelian group homomorphism M −!M given by m 7! g2G=H gm. Here the notation g 2 G=H is short for \take a set of coset representatives"; the choice of H representatives is immaterial since m 2 M . If we use a projective resolution P·Z to compute group cohomology, τ is induced by the chain map H G HomZH (P·;M) = (HomZ(P·;M)) −!(HomZ(P·;M)) = HomZG(P·;M): P again using the operator g2G=H g. The following key fact is then obvious: Proposition 2.1 τ(H; G) ◦ i∗(H; G) is multiplication by [G : H]. Taking H to be the trivial group, we obtain: n n Corollary 2.2 jGj · H (G; M) = 0 for all n > 0. In particular, H (G; Q) = 0 for all n > 0. Since the rational cohomology is trivial, we will focus on mod p cohomology for p prime. ∗ ∗ Corollary 2.3 If [G : H] is prime to p, H (G; Fp)−!H (H; Fp) is injective. In particular, n (i) this is true for H the p-Sylow subgroup of G, and (ii) if jGj is prime to p then H (G; Fp) = 0 for all n > 0. In the next corollary we again take Fp-coefficients. Corollary 2.4 Suppose i(H; G)∗ is onto and p divides [G : H]. Then τ(H; G) is identically zero. In particular, if G is a p-torus and H is a proper subgroup then τ(H; G) = 0. The proposition itself is actually a corollary of the stronger assertion: 3 Proposition 2.5 Taking coefficients in a commutative ring R, τ(H; G) is a map of H∗(G; R)- modules: τ(ai∗b) = (τ(a))b: Proof: We use the singular chain complex C∗EG as our free resolution of R. Using the standard cochain formula for the cup product, we see that the (non-commutative) differential graded algebra C∗EG is a G-algebra; i.e. g · (xy) = (g · x)(g · y). It is then immediate that the cochain level transfer (C∗EG)H −!(C∗EG)G defined above is a map of both left and right C∗BG = (C∗EG)G-modules. We turn next to the double coset formula, which concerns composition in the reverse order: transfer followed by restriction. It looks somewhat complicated at first glance, but in fact it is easy to prove and easy to use. Although there is a more conceptual explanation of the formula that we'll give later, the explicit form given here is very useful. Suppose K; H ⊂ G, and consider the double cosets KnG=H, thought of as the K-orbits of the natural K-action on G=H. Proposition 2.6 X −1 −1 −1 ∗ i(K; G) ◦ τ(H; G) = τ(K \ xHx ;K) ◦ i(K \ xHx ; xHx ) ◦ cx−1 ; x2KnG=H where the sum is over a set of double-coset representatives. Proof: We work on the cochain level, using injective resolutions. The lefthand side is just H τ G K P the composite I −! I −!I , where τα = g2G=H gα. A typical term on the right is given by the composite IH −!x IxHx−1 ⊂ IK\xHx−1 −!τx IK ; P −1 where τx(α) = k kα and the sum is over k 2 K=(K \ xHx ). Hence the composite is given by α 7! P(kx)α: But K \ xHx−1 is the isotropy group in K of xH. So in this last sum kx is ranging over the elements of the orbit KxH=H.

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