Hatcher-Algebraic-Topology · Problem 1.3.15

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

Let

p : \tilde{X} \to X

be a simply-connected covering space of

X

and let

A \subset X

be a path-connected, locally path-connected subspace, with

\tilde{A} \subset \tilde{X}

a path-component of

p^{-1}(A)

. Show that

p : \tilde{A} \to A

is the covering space corresponding to the kernel of the map

\pi_1(A) \to \pi_1(X)