I will present joint work with Madsen, Tillmann, Weiss (arXiv: math/0605249). The d-dimensional cobordism category has as objects all closed (d-1)-manifolds, and as morphisms all compact d-dimensional cobordisms. To any category C one can associate a classifying space BC, built by first taking a point for each object of C, then attaching an edge between x and y for each morphism from x to y, then attaching a 2-simplex for each commutative triangle, and so on. In general, the set of connected components of BC is the set of objects modulo the equivalence relation generated by the morphisms. When C is the cobordism category, this gives the set of cobordism classes of (d-1)-manifolds, as calculated by Thom. In this talk I will explain how to completely determine the homotopy type of BC. As a corollary we get a new proof of Madsen-Weiss' generalized Mumford conjecture. |