Hatcher-Algebraic-Topology · Problem 4.1.4

Homotopy Theory

Mathematics — Graph Theory · universal-cover-homotopy-groups-action

Problem

Let

p : \widetilde{X} \to X

be the universal cover of a path-connected space

X

. Show that under the isomorphism

\pi_n(X) \approx \pi_n(\widetilde{X})

, which holds for

n \geq 2

, the action of

\pi_1(X)

\pi_n(X)

corresponds to the action of

\pi_1(X)

\pi_n(\widetilde{X})

induced by the action of

\pi_1(X)

\widetilde{X}

as deck transformations. More precisely, prove a formula like

\gamma p_*(\alpha) = p_*\bigl(\beta_{\tilde{\gamma}}(\gamma_*(\alpha))\bigr)

where

\gamma \in \pi_1(X, x_0)

\alpha \in \pi_n(\widetilde{X}, \tilde{x}_0)

, and

\gamma_*

denotes the homomorphism induced by the action of

\gamma

\widetilde{X}