L3

It's well known that the mapping space of two finite dimensional

manifolds can be given the structure of an infinite dimensional manifold

modelled on Frechet spaces (provided the source is compact). However, it is

not that the charts on the original manifolds give the charts on the mapping

space: it is a little bit more complicated than that. These complications

become important when one extends this construction, either to spaces more

general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to

transport local properties of a space to local properties of its mapping

space. As an application, we shall show that applying the mapping

construction to a regular map is again regular.

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.