Algebra/Topology seminar

Speaker: Nathan Perlmutter (University of Oregon)

Title: Homological stability for the diffeomorphism groups of odd dimensional manifolds

Abstract:

Let $M$ be a compact, smooth manifold and let $\Diff(M)$ denote the group of diffeomorphisms of $M$, topologized in the $C^{\infty}$ topology.

In the case that $\dim(M) = 2n \geq 6$, recent results of Galatius and Randal-Williams identify the homological type of the classifying spaces $\BDiff(M)$ upon ``stabilizing'' the underlying manifold $M$, with respect to forming the connected sum with copies of $S^{n}\times S^{n}$.Now consider the case when $\dim(M)$ is odd. I will present some new homological stability theorems for the diffeomorphism groups of manifolds of dimension $2n+1 \geq 9$, with respect to forming the connected sum with copies of $S^{n}\times S^{n+1}$ and with other $(n-1)$-connected, $(2n+1)$-dimensional manifolds.