From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Parallelizabilty of 3-manifolds
Date: 17 Jul 1999 06:56:26 -0400
Newsgroups: sci.math

strohmai@physik.uni-leipzig.de (Alexander Strohmaier) writes:

>In Steenrod, The Topology of Fibre Bundles, 1951
>I read the statement that orientable 3 manifolds
>are parallelizable.
>There an old paper of Stiefel is cited, with
>the remark that the arguments are sketchy, but
>likely to be correct (??).
>Does anyone know about a modern argument for
>this?

They abound, but the various arguments have various 
prerequisites (in knowledge about 3-manifolds in 
particular, topology in general, or both).  What
are you willing to assume about orientable 
3-manifolds?  That they have open-book structures?
That they can be constructed from the 3-sphere by
surgery?

I would guess that Stiefel's argument was by (a
sort of what later became) obstruction theory,
and that (if so, maybe even if not) it would be
easy nowadays for lots of people to remove the
sketchiness (because of better, and more widely
available, knowledge of various things that have
become elementary since Stiefel's day; for instance,
I bet that what "everyone" now knows about homotopy
classes of maps form S^3 to S^2 can be applied to
remove some of his sketchiness).  Your name and
domain suggest that you're more likely than I to
be able to find and read Stiefel's original paper--
why don't you do so, and come back to sci.math with
a report on it?
 
Lee Rudolph