How Does Group Action Determine the Structure of Covering Spaces in Topology?

[harbottle]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

See above

 

[mighty2000]

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Thank you for reaching out for help with this problem from Hatcher. I understand that you are having trouble with the given problem and are seeking guidance. my expertise lies in analyzing and solving problems through the scientific method. I will do my best to assist you with this problem.

Firstly, it is important to note that a covering space action of a group G on a path-connected, locally path-connected space X can be defined as a continuous map p: X -> X/G, where X/G is the quotient space of X by the action of G. This map is called the projection map.

Now, let's address the three parts of the problem:

a. To show that every path-connected covering space between X and X/G is isomorphic to X/H for some subgroup H in G, we can use the notion of the fundamental group. Since X and X/G are path-connected and locally path-connected, their fundamental groups are isomorphic. This means that π1(X) ≅ π1(X/G). Now, by the Galois correspondence, there is a one-to-one correspondence between the subgroups of π1(X) and the covering spaces of X up to isomorphism. This means that for every subgroup H in G, there exists a corresponding covering space X/H of X. Since X and X/G have isomorphic fundamental groups, there must exist a subgroup H in G such that X/H is isomorphic to X/G.

b. To show that two covering spaces X/H1 and X/H2 of X/G are isomorphic iff H1 and H2 are conjugate subgroups of G, we can use the notion of deck transformations. A deck transformation of a covering space X/H is a homeomorphism of X that preserves the fibers of the covering map p: X -> X/H. This means that for every x in X, p(φ(x)) = p(x) for all deck transformations φ. Now, if X/H1 and X/H2 are isomorphic, then there exists a deck transformation φ such that φ(X/H1) = X/H2. This implies that H1 and H2 are conjugate subgroups of G, since φ is a homeomorphism that preserves the fibers of p.

c. To show that the covering space X/H -> X/G is normal iff H is a normal subgroup of G, we can use the notion of normal covering spaces. A covering space X/H