Hatcher-Algebraic-Topology · Problem 3E.1

Cohomology

Mathematics — Graph Theory · bockstein-homomorphisms

Problem

Show that H∗(K(Zm,1);Zk)H^*(K(\mathbb{Z}_m, 1); \mathbb{Z}_k) is isomorphic as a ring to H∗(K(Zm,1);Zm)⊗ZkH^*(K(\mathbb{Z}_m, 1); \mathbb{Z}_m) \otimes \mathbb{Z}_k if kk divides mm. In particular, if m/km/k is even, this is ΛZk[x]⊗Zk[y]\Lambda_{\mathbb{Z}_k}[x] \otimes \mathbb{Z}_k[y].

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