# Homework Help: Covering space action

1. Oct 12, 2009

### harbottle

1. The problem statement, all variables and given/known data

I am having trouble with this problem from Hatcher:

24. Given a covering space action of a group G on a path-connected, locally path-connected space X, then each subgroup H in G determines a composition of covering spaces X -> X/H -> G. Show:

a. Every path-connected covering space between X and X/G is isomorphic to X/H for some subgroup H in G

(The best I can do here is say that since we have a covering space action then pi_1(X/H1) = pi_1(X/H2); not sure how to proceed.)

b. Two such covering spaces X/H1 and H/H2 of X/G are isomorphic iff H1 and H2 are conjugate subgroups of G.

c. The covering space X/H -> X/G is normal iff H is a normal subgroup of G, in which case the group of deck transformations of this cover is G/H.

Any ideas? The two preceding propositions look tantalisingly close to what I need but I can't massage the problem to accommodate them.

2. Relevant equations

Page 71-3 in this pdf http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf)

3. The attempt at a solution

See above