Re: covering space

From: W. Dale Hall (mailtowd-hall_at_pacbell.net)
Date: 09/08/04 

Date: Wed, 08 Sep 2004 17:47:28 GMT

alliville wrote:
> A book gives the following example about covering that I don't quite
> undertand.
> It says: X=polyhedron, A=subpolyhedron. p: M-->X be a universal covering
> of X and B=path component of p^(-1)(A) containing the equivalence class
> of the constant path at x_0 in A.
> Let p' be the restriction of p to B.
> It says: p': B--> A is not a universal covering of A in general (why?)
> But it says p': B-->A is the covering space of the kernel of
> i_*: pi_1 (A, x_0)-->pi_1(X,x_0)
> [Here: pi_1(A,x_0) means fundamental group etc and i_* is the induced
> map from the inclusion map]
> (What does it mean? And why is it true?)
>

This is a situation in which a trivial example is illustrative. The fact
that the author is discussing polyhedra is not really important, except
for the facts that (1) he's only developing techniques for polyhedra so
far, and (2) the setting may allow him to avoid some pathological cases
that the general setting presents. For this particular problem, it's not
too important to keep polyhedra in mind.

With that in mind, take X = a nice space that's already 1-connected, and
then A = some nice subspace that's connected, but not simply-connected.

Now look at what the author is saying.

Dale