Newsgroups: sci.math
From: kwhyte@hog.uchicago.edu (Kevin Whyte)
Subject: Non-paracompact manifolds
Date: Wed, 1 Apr 1992 06:04:49 GMT


  Let me give some motivation, since the question I want
to ask is very vague.  I recently ran in to the long line
again, after almost forgetting about it since point-set
topology.  It is a non-paracompact manifold with some strange
properies ( The tangent bundle is non-trivial, it has inf.
many distinct analytic structures, etc.).

 Brief review of the long line:

 First, the idea is to stick uncountably many copies of [0,1)
end to end.  The real line is countably many, i.e. take the
set [0,1) x N with the topology coming from the dictionary
ordering.  This is just [0,inf) in the obvious way.  To do
the same thing uncountably many times, take S to be the minimal,
uncountable, well-ordered set (minimal meaning for all s in S
the set of t<s is countable).  Then we simply take [0,1) x S
with the dictionary topology, and we have the long ray.  Removing
the first 0 gives the long line.

End of review.

 To me, this looked like someone was trying to construct the
universal cover of the circle by sticking fundamental domains
together, but they attached far too many (and no, the obvious
map from the long line to the circle is not continuous since,
in fact, any continuous map from the long line to the reals
(or any paracompact manifold) is eventually constant!). So this
led me to try to do the same with other manifolds.  Thinking
of the universal cover as one fundamental domain per element of
the fundamental group stuck together by the generators, all we 
need is an uncountable set with injective maps corresponding to
mult. by the generators of the fund. group satisfying the 
appropriate relations.  Here lies the difficulty: does such a 
thing always exist? And what should minimality mean here - which
sets should be forced to be countable?  Any ideas?  Of course,
no one really cares about non-paracompact manifolds anyway.

Kevin
kwhyte@math.uchicago.edu

ps: Does anyone know any slick examples of a compact, simply
connected smooth manifold not of the form G/H?