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 tr_{H}^{G} : H_{1}(G; ℤ) → H_{1}(H; ℤ) as a homomorphism G_{ab} → H_{ab} on the abelianizations of G and H.

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

Exercise 5.3. Let S_{3} be the group of permutations of a set with three elements. Let ℤ have the trivial S_{3}-action. Compute

H^{∗}(S_{3}; ℤ)

using subgroups and invariants.

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

colim_{I} H_{∗}(G_{i}; M) → H_{∗}(colim_{I} G_{i}; 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

tr_{H}^{G} ((res_{H}^{G} u) ∪ v) = u ∪ (tr_{H}^{G} v).

Exercise 5.6. Let n ∈ ℕ be an even natural number and C_{n} be the cyclic group of order n. Compute H^{∗}(C_{n}; 𝔽_{2}) and H_{∗}(C_{n}; 𝔽_{2}) as 𝔽_{2}-algebras, with the cup and Pontrjagin products, respectively.

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