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COHOMOLOGY OF GROUPS : EXERCISES Sheet 5, 4 June 2007

Exercise 5.1. Let G be a group, H be a subgroup of finite index, and let Z have trivial G-action. Give a formula for the transfer trGH : H1(G;Z) → H1(H;Z) as a homomorphism Gab →Hab on the abelianizations of G and H.

Exercise 5.2. Write a detailled proof of the Cartan-Eilenberg double coset for- mula, using the hints given in the lecture.

Exercise 5.3. Let S3 be the group of permutations of a set with three elements.

Let Z have the trivial S3-action. Compute H∗(S3;Z) using subgroups and invariants.

Exercise 5.4. Prove that if {Gi}i∈I is a directed system of groups, then for any Z[G]-module M the natural map

colim

I H∗(Gi;M)→H∗(colim

I Gi;M) is an isomorphism.

Exercise 5.5. Let G be a group and H be a subgroup of finite index. Prove the transfer formula for the cup-product in cohomology given in the lecture, namely

trGH (resGHu)∪v

=u∪(trGHv).

Exercise 5.6. Let n ∈N be an even natural number and Cn be the cyclic group of order n. ComputeH∗(Cn;F2) andH∗(Cn;F2) as F2-algebras, with the cup and Pontrjagin products, respectively.

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