Homological stability of diffeomorphism groups

TL;DR

This paper establishes a stability theorem for block diffeomorphisms of certain 2d-dimensional manifolds, enabling the computation of their classifying space homology in a stable range by combining recent theorems and lemmas.

Contribution

It proves a new stability theorem for block diffeomorphisms of connected sums of S^d x S^d, advancing understanding of their homology in a stable range.

Findings

Homology of classifying spaces determined in stable range

Stability theorem for block diffeomorphisms proved

Integration of recent theorems and lemmas for homology computation

Abstract

In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range.