Homology stability for symmetric diffeomorphism and mapping class groups

TL;DR

This paper proves homology stability for classifying spaces of diffeomorphism groups fixing points or disks, and for symmetric diffeomorphisms of connected sums, extending to mapping class groups and other cases.

Contribution

It establishes homology stability results for symmetric diffeomorphism and mapping class groups of manifolds, including new cases involving connected sums with multiple copies of a manifold.

Findings

Homology stability for diffeomorphism groups fixing points or disks.

Homology stability for symmetric diffeomorphisms of connected sums.

Extensions to mapping class groups and broader generalizations.

Abstract

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of $W$ connected sum with $k$ copies of an arbitrary compact smooth manifold $Q$ of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.