COHOMOLOGY OF GROUPS : EXERCISES

Sheet 4, 14 Mai 2007

Exercise 4.1. Let n ⩾ 2 be an integer, let G = C_{p^{n}} = ⟨ t | t^{p^{n}} = 1 ⟩ be the cyclic group of order p^{n}, and let H < G be the subgroup generated by t^{p}, so that H ≅ C_{p^{n−1}}. Compute

cor_{H}^{G} : H_{∗}(H, M) → H_{∗}(G, M) and tr_{H}^{G} : H_{∗}(G, M) → H_{∗}(H, M)

for M = 𝔽_{p} or M = ℤ with the trivial G-action. Compute

tr_{H}^{G} : H^{∗}(H, M) → H^{∗}(G, M) and res_{H}^{G} : H^{∗}(G, M) → H^{∗}(H, M)

for the same values of G, H and M.

Exercise 4.2. Let G be a finite group, H be a subgroup of G, and let K be any group. Let 𝔽 be a field with trivial action of G and K, and consider the Künneth isomorphism

κ : H^{∗}(G; 𝔽) ⊗_{𝔽} H^{∗}(K; 𝔽) ⟶^{≅} H^{∗}(G × K; 𝔽).

Show that res_{H×K}^{G×K} κ = κ(res_{H}^{G} ⊗ id), and tr_{H×K}^{G×K} κ = κ(tr_{H}^{G} ⊗ id). Repeat the exercise in homology, for cor_{H×K}^{G×K} and tr_{H×K}^{G×K} (in homology, the hypothesis that G be finite can be supressed, but of course we need [G : H] < ∞ for the transfer to be defined).

Exercise 4.3. Repeat exercise 4.1 for G = C_{p}^{n} and H a subgroup of order p^{n−1}.

Exercise 4.4. Let G be a group, x ∈ G, and let τ_{x} : G → G be defined by g ↦ xgx^{−1}. Show that Bτ_{x}, as a self map of BG, is homotopic to the identity.

Exercise 4.5. Let G be a group, M be a ℤG-module, and take x ∈ G. Show that the homomorphism c : H_{∗}(G; M) → H_{∗}(G; M) induced by τ_{x} : G → G and M → M, m ↦ xm, is equal to the identity.

Exercise 4.6. Let K ⊂ L be a finite Galois extension of fields with Galois group G. Show that

H^{n}(G; L) ≅ H_{n}(G; L) ≅ \{ \begin{matrix}K & if n = 0, \\ 0 & if n ≠ 0. \end{matrix}

Hint: use the normal basis theorem.

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