Algebraic Topology · Problem 3E.1

Cohomology — Allen Hatcher

Mathematics — Topology · bockstein-cohomology-ring-Km1

Problem

Show that

H^*(K(\mathbb{Z}_m, 1); \mathbb{Z}_k)

is isomorphic as a ring to

H^*(K(\mathbb{Z}_m, 1); \mathbb{Z}_m) \otimes \mathbb{Z}_k

k

divides

m

. In particular, if

m/k

is even, this is

\Lambda_{\mathbb{Z}_k}[x] \otimes \mathbb{Z}_k[y]