From: ctm@bosco.berkeley.edu (C. T. McMullen)
Newsgroups: sci.math.research
Subject: Re: PL and DIFF manifolds: a question
Date: 21 Aug 1997 18:26:12 GMT

In article <Pine.SUN.3.96.970819234227.25484B-100000@stlawrence.maths>,
Marco de Innocentis  <mdi@NOSPAMmaths.ox.ac.uk> wrote:
>
> It is often stated
>that DIFF manifolds are a particular case of PL manifolds, but this fact
>is not obvious. How does one go about proving it?

A DIFF manifold carries a canonical PL structure, which can
be obtained by 
	1) introducing a Riemannian metric, then
	2) choosing a fine triangulation whose 1-skeleton is piecewise
	geodesic, with triangles of bounded geometry; and
	3) using barycentric coordinates (relative to the given metric)
	to get new, nearby simplices, each endowed with a
	linear structure.
It is not quite true that a DIFF manifold *is* a PL manifold;
one should introduce the intermediate category of
PDIFF, piecewise-smooth manifolds, which clearly contains DIFF
manifolds.  A clear discussion appears in the book by
Thurston, "Three-Dimensional Geometry and Topology", PUP, 1997.
	-C. McMullen