Stable moduli spaces of high dimensional manifolds

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.