From: Dan Christensen <jdc@scratchy.net.dhis.org>
Subject: Re: Algebraic topology for physicists
Date: 8 Nov 2000 10:13:35 GMT
Newsgroups: sci.physics.research
Summary: [missing]

baez@galaxy.ucr.edu (John Baez) writes:
> penny314@aol.com (Penny314) wrote:
> >Does anyone know how to do obstruction theory if the space X is simple
> >(the action of pi-1 on each p1-n is trivial) and a reference for that?
> >                              [she means non-trivial]
> 
> But I'm still annoyed that none of the homotopy theory gods stepped in and 
> explained the subtleties that arise when you're using a Postnikov tower to
> describe a non-simple space.  I think this is discussed in Peter May's cute
> little book on "Simplicial Objects in Algebraic Topology", but I need to go
> to the library and get ahold of it.   

May's book is one of the references I checked when I looked this up.
The relevant theorem there is 25.7 on page 113, where he makes the even
stronger assumption that there are no maps from pi_1 to Aut(pi_n).  In
Remark 25.9 (2), he says that this can be weakened to assuming that
the usual action of pi_1 on pi_n is trivial (n-simplicity).  Then he
says:  "Further, the appropriate generalization to the case where
n-simplicity fails could be developed by use of the results on the
structure of A(K(pi,n))'s."  (Here A(X) is the simplicial group of
automorphisms of X.)  But he doesn't give any references.

Unfortunately, I don't know a good explanation of why non-trivial
actions give such trouble.  Maybe it is because cohomology with
twisted coefficients isn't representable as homotopy classes of maps
to an Eilenberg-Mac Lane space, so the fibrations occuring the in
Postnikov tower won't be the homotopy fibres of maps between spaces?

When you are doing obstruction theory, part of the problem is that the
fibres of a fibre bundle don't have canonical basepoints, and keeping
track of choices of basepoints is a real pain.  It makes me wonder
if we should be using groupoids in such situations.

Dan