Algebraic Topology · Problem 4I.2

Homotopy Theory — Allen Hatcher

Mathematics — Topology · stable-splittings-of-spaces

Problem

Using the K\"unneth formula, show that

\Sigma K(\mathbb{Z}_m \times \mathbb{Z}_n, 1) \simeq \Sigma K(\mathbb{Z}_m, 1) \vee \Sigma K(\mathbb{Z}_n, 1)

m

and

n

are relatively prime. Thus to determine stable splittings of

K(\mathbb{Z}_n, 1)

it suffices to do the case that

n

is a prime power, as in Proposition 4I.3.