A finite combinatorial presentation for closed smooth manifolds

by Prof. L.C. Siebenmann (U. Paris-Sud, Orsay, France)

Wednesday 10 Jul 2013, 02:30 → 03:30 Asia/Kolkata

AG-69 (Colaba Campus)

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.