Algebraic Topology · Problem 26

Homotopy Theory — Allen Hatcher

Mathematics — Topology · isomorphic-homotopy-groups-not-homotopy-equivalent

Problem

Generalizing the example of

\mathbb{R}P^2

and

S^2 \times \mathbb{R}P^\infty

, show that if

X

is a connected finite-dimensional CW complex with universal cover

\tilde{X}

, then

X

and

\tilde{X} \times K(\pi_1(X), 1)

have isomorphic homotopy groups but are not homotopy equivalent if

\pi_1(X)

contains elements of finite order.