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 $\mathbb R^{n+s}$ such that, by a well defined smoothing process, $K$ inherits from $\mathbb R^{n+s}$ a smooth submanifold structure that is well defined up to concordance in the sense of $M$. Hirsch. Every closed smooth $n$-submanifold of $\mathbb R^{n+s}$ is so presented. Ideas of S. Cairns and J.H.C. Whitehead are used.
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 $\mathbb R^{n+s}$ such that, by a well defined smoothing process, $K$ inherits from $\mathbb R^{n+s}$ a smooth submanifold structure that is well defined up to concordance in the sense of $M$. Hirsch. Every closed smooth $n$-submanifold of $\mathbb R^{n+s}$ 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.