Date: Tue, 8 Dec 1998 11:08:12 -0800 (PST)
From: Nick Halloway <snowe@rain.org>
Subject: Weird covering spaces


Hi ...
I'm learning algebraic topology on my own, and
I'm looking for some insight into non-normal covering spaces -- what would
block the group of covering transformations from being transitive.

I found a non-normal 3-fold cover for a Klein bottle.  If the Klein
bottle is made by identifying a square like so:

_____a>______
|           |
b           b
v           v
|           |
|____<a_____|

then the homotopy group is <a,b | b^(-1)ab = a^(-1)> and a
non-normal subgroup of index 3 is generated by a^3 and b.  A non-normal
3-fold covering space corresponds to this non-normal subgroup.  The
3-fold covering space is also a Klein bottle.

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?

ALSO, if you have a space S with covering space C, then the homotopy
group pi_1(C,c_0) injects into pi_1(S,s_0).  You can represent pi_1(S,s_0)
as permutations on the sheets of C over S -- if c in C is projected onto
s_0, then for g in pi_1(S,s_0), define a permutation s(g) by
s(g)(c) = c', where c' is the ending point of loop g when it is lifted to
C so that it starts at c.  The kernel of this representation is the
largest normal subgroup N of pi_1(S,s_0) contained in pi_1(C,c_0).

What does the group pi_1(S,s_0)/N tell you about the type of covering
space that C is?  For the 3-fold Klein bottle cover above,
pi_1(S,s_0)/N is S_3.

It would determine the group G of covering transformations of (C,c_0)
over (S,s_0), since G is isomorphic to
Norm(pi_1(C,c_0) in pi_1(S,s_0)) / pi_1(C,c_0)
and that is independent of factoring out normal subgroups contained
in pi_1(C,c_0).  Anything else?

for example you can have 3-fold covers for a Klein bottle that have
pi_1(S,s_0)/N isomorphic to C_3 or to S_3.  Do these tell you anything
about the knottedness of an embedding of the Klein bottle in 4-space?
and so on.