Homological stability of diffeomorphism groups of high dimensional manifolds via $E_k$-algebras

TL;DR

This paper investigates the homological stability of high-dimensional manifold diffeomorphism groups using $E_k$-algebras, introducing improved stability bounds and a novel quantised stability concept.

Contribution

It develops new stability results for diffeomorphism groups of certain high-dimensional manifolds, especially with rational coefficients, and introduces the concept of quantised homological stability.

Findings

Improved stability bounds with rational coefficients.

Introduction of quantised homological stability.

Establishment of linear and at least 2/3 slope stability results.

Abstract

We will study homological stability of the diffeomorphism groups of the manifolds $W_{g,1}:=D^{2n} \# (S^n \times S^n)^{\#g }$ using $E_k$-algebras. This will lead to new improvements in the stability results, especially when working with rational coefficients. Moreover, we will prove a new type of stability result -- quantised homological stability -- which says that either the best stability result is a linear bound of slope $1/2$ or the stability is at least as good as a line of slope $2/3$.