From: Greg Kuperberg <greg@math.ucdavis.edu>
Subject: Re: weird covering spaces
Date: Thu, 17 Dec 1998 11:43:09 -0800 (PST)

> I can see why this covering space doesn't have a transitive group
> of covering transformations.  Actually the group of covering
> transformations only has the identity.  But could anyone give more
> insight as to what other reasons there would be for the group of
> covering transformations not being transitive?

The point is this.  Support that a group G is the fundamental group
of some nice space A and that X is the universal cover.  G acts freely
on X.  If H is a subgroup of G, then H also acts freely and X/H is an
intermediate cover of A.  But which elements of G preserve the H-orbits?
If you have an H-orbit Hx, then gHx is the gHg^{-1} - orbit of gx.
This is an H-orbit if and only if g normalizes H.  This tells you
precisely the extent to which transitivity fails.  Given two points x
and gx in X lying over the same point in A, if g does not normalize H
then no element of G sends Hx to Hgx.

The largest normal subgroup of H is sort-of the other end of things.
It's a way to repair an irregular cover to get a regular one.  In fact
there is an intrinsic description:  If pi:Y -> A is an intermediate
covering map corresponding to the subgroup H, make a new bundle Z
over A whose fiber at a is the set of all linear orderings of the points
in the fiber pi^{-1}(a).  Then a connected component of Z is a regular
covering whose fundamental group is the largest normal subgroup of H.

                        Greg