Nathan Perlmutter, Stanford University

Title: Moduli Spaces of Manifolds

Abstract: Let M be a smooth manifold, let Diff(M) denote the topological group of self-diffeomorphisms of M and let BDiff(M) denote the classifying space. For any paracompact space X, there is a one-one correspondence between the set of homotopy classes [X, BDiff(M)], and the set of isomorphism classes of smooth fibre bundles over X with fibre M. The classifying space BDiff(M) is referred to as the moduli space of manifolds of type M. The study of the homotopy type of these spaces occupies a central place in smooth topology.

In this talk I will discuss some contemporary approaches to studying the homotopy type of BDiff(M), for varying M. In particular I will discuss the work of Madsen and Weiss identifying the homological type of the moduli spaces of Riemann surfaces and the results of Galatius and Randal-Williams on the moduli spaces of manifolds of dimension 2n. I will then present recent work of mine pertaining to the moduli spaces of odd dimensional manifolds, and manifolds with boundary, and will discuss connections to cobordism categories and surgery theory.