Stable moduli spaces of high dimensional manifolds

29 October 2012
Oscar Randal-Williams
I will discuss recent joint work with S. Galatius, in which we generalise the Madsen--Weiss theorem from the case of surfaces to the case of manifolds of higher even dimension (except 4). In the simplest case, we study the topological group $\mathcal{D}_g$ of diffeomorphisms of the manifold $\#^g S^n \times S^n$ which fix a disc. We have two main results: firstly, a homology stability theorem---analogous to Harer's stability theorem for the homology of mapping class groups---which says that the homology groups $H_i(B\mathcal{D}_g)$ are independent of $g$ for $2i \leq g-4$. Secondly, an identification of the stable homology $H_*(B\mathcal{D}_\infty)$ with the homology of a certain explicitly described infinite loop space---analogous to the Madsen--Weiss theorem. Together, these give an explicit calculation of the ring $H^*(B\mathcal{D}_g;\mathbb{Q})$ in the stable range, as a polynomial algebra on certain explicitly described generators.