Colloquium abstracts

L.C. Siebenmann
U. Paris-Sud, Orsay, France
July 10, 2013

A finite combinatorial presentation for closed smooth manifolds:  I will define a class of finite simplicial n-complexes K simplexwise linearly embedded in $\R^{n+s}$ such that, by a well defined smoothing process, K inherits from $\R^{n+s}$ a smooth submanifold structure that is well defined up to concordance in the sense of M. Hirsch. Every closed smooth n-submanifold of $\R^{n+s}$ is so presented. Ideas of S. Cairns and J.H.C. Whitehead are used. In 1991, Macpherson conjectured a quite different finite combinatorial presentation for closed smooth manifolds; it involves matroids. But the basic question whether it really determines a smooth structure up to diffeomorphism or concordance is (I believe) still open.