Past Geometry and Analysis Seminar

Open Riemann surfaces and the Weil-Petersson Poisson structure

Let M be a closed spin manifold.

Gromov and Lawson have shown that the presence of certain "enlargeable"

submanifolds of codimension 2 is an obstruction to the existence of a Riemannian metric with positive scalar curvature on M.

In joint work with Hanke, we refine the geoemtric condition of

"enlargeability": it suffices that a K-theoretic index obstruction of the submanifold doesn't vanish.

A "folk conjecture" asserts that all index type obstructions to positive scalar curvature should be read off from the corresponding index for the ambient manifold M (this this is equivalent to a small part of the strong Novikov conjecture). We address this question for the obstruction above and discuss partial results.

Abstract:

(joint work with Allen Hatcher) Let M be a compact, connected 3-manifold with a

fixed boundary component d_0M. For each prime manifold P, we consider the

mapping class group of the manifold M_n^P obtained from M by taking a connected

sum with n copies of P. We prove that the ith homology of this mapping class

group is independent of n in the range n>2i+1. Our theorem moreover applies to

certain subgroups of the mapping class group and include, as special cases,

homological stability for the automorphism groups of free groups and of other

free products, for the symmetric groups and for wreath products with symmetric

groups.