Algebraic Topology · Problem 32

The Fundamental Group — Allen Hatcher

Mathematics — Topology · covering-spaces

Problem

Consider covering spaces

p : \tilde{X} \to X

with

\tilde{X}

and

X

connected CW complexes, the cells of

\tilde{X}

projecting homeomorphically onto cells of

X

. Restricting

p

to the 1-skeleton then gives a covering space

\tilde{X}^1 \to X^1

over the 1-skeleton of

X

. Show: (a) Two such covering spaces

\tilde{X}_1 \to X

and

\tilde{X}_2 \to X

are isomorphic iff the restrictions

\tilde{X}_1^1 \to X^1

and

\tilde{X}_2^1 \to X^1

are isomorphic. (b)

\tilde{X} \to X

is a normal covering space iff

\tilde{X}^1 \to X^1

is normal. (c) The groups of deck transformations of the coverings

\tilde{X} \to X

and

\tilde{X}^1 \to X^1

are isomorphic, via the restriction map.