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Lectures on the Cohomology of Groups

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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2. (a) If X is a point, then H∗G (X, M ) = H∗ (G, M ). More generally, this holds if X is contractible. Thus equivariant homology is the same as the homology of the group if the space is trivial. [Sketch of proof: There is a weak equivalence C(X, M ) → M , where M is viewed as a chain complex concentrated in dimension 0, and a weak equivalence is a map that induces an isomorphism in homology. ] (b) At the other extreme, if G is the trivial group, then H∗G (X, M ) = H∗ (X, M ). (c) If G acts freely on X, and Y := G\X, then H∗G (X, M ) = H∗ (Y, M ); here M on the right side is viewed as a π1 (Y )-module (and hence a local coefficient system on Y ) via the canonical map π1 (Y ) → G provided by the theory of covering spaces.

In summary, there are two spectral sequences converging to H∗ (T C), based on the two viewpoints (a) and (b). ] 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.

Iii) For every prime p, every elementary abelian p-subgroup of G has rank at most 1. (iv) The Sylow subgroups of G are cyclic or generalized quaternion groups. The condition that G have periodic cohomology is very restrictive, and the groups with this property have been completely classified. A less restrictive (but still quite useful) condition is periodicity of the p-primary component H ∗ (G)(p) for a fixed prime p. More briefly, we say that G has p-periodic cohomology. There are various characterizations of this property analogous to the results stated above.