$E_2$-cells and mapping class groups

TL;DR

This paper introduces a new form of homological stability called 'secondary homological stability' for mapping class groups of orientable surfaces, achieved through CW approximations in the category of $E_2$-algebras.

Contribution

It develops a novel stabilization result and constructs CW approximations in $E_2$-algebras with controlled cell structures, advancing understanding of mapping class group homology.

Findings

Proves secondary homological stability for mapping class groups.

Constructs CW approximations with no $E_2$-cells below a certain line.

Establishes new techniques in algebraic topology for surface groups.

Abstract

We prove a new kind of stabilisation result, "secondary homological stability", for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_2$-algebras, which have no $E_2$-cells below a certain vanishing line.