From: Greg Kuperberg
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