By Kenneth S. Brown
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Cohomology of teams and algebraic K-theory, 131–166, Adv. Lect. Math. (ALM), 12, Int. Press, Somerville, MA, 2010, model 18 Jun 2008
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We have introduced these spectral sequences because they are useful in connection with equivariant homology, to which we turn next. 2 Equivariant homology Equivariant homology is the same as what L¨ uck calls “Borel homology” in his lectures in this volume, but I will describe an algebraic approach. For simplicity I will stick to homology, but everything I say has an analogue for cohomology. 1. If X is a G-CW-complex and M is a G-module, then we set H∗G (X, M ) := H∗ (P ⊗G C(X, M )), where P is a projective resolution of Z over ZG and C(X, M ) is the cellular chain complex of X with coefficients in M (with diagonal Gaction).
The second version of the result is more concise. It simply says that the canonical map H(G)(p) → lim H(P ) P is an isomorphism in positive dimensions, where the limit (or inverse limit) is taken over the category whose objects are the p-subgroups P of G and whose morphisms are the maps P1 → P2 induced by conjugation by elements of G. 1. The proof of the theorem is based on formal properties of the restriction and corestriction maps between H(G) and H(S). Recall first that the composite H(G) → H(S) → H(G) (restriction followed by corestriction) is simply multiplication by the index [G : S], which is relatively prime to p.
Thus we always have maps in both directions, and formal differences between homology and cohomology disappear. 9). 3): H(H, M ) ∼ = H(G, IndG H M ), with no distinction between homology and cohomology. 2 Local computation of homology and cohomology We continue to write H(G) for homology or cohomology with an arbitrary coefficient module. 4, which states that H(G) is annihilated by |G| in positive dimensions. 1) p in positive dimensions, where p ranges over the primes dividing |G|, and (−)(p) denotes the p-primary component.
Lectures on the Cohomology of Groups by Kenneth S. Brown
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