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 ,
Marco de Innocentis 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