Secondary homological stability for mapping class groups of nonorientable surfaces

TL;DR

This paper proves a secondary homological stability theorem for nonorientable surface mapping class groups using cellular $E_2$-algebras, leading to improved stability ranges for their homology.

Contribution

It introduces a new secondary stability theorem for nonorientable surface mapping class groups utilizing the Galatius--Kupers--Randal-Williams framework.

Findings

Established a new stability range for homology of nonorientable surface mapping class groups.

Applied cellular $E_2$-algebras to prove secondary stability.

Improved understanding of homological properties with added torus holes.

Abstract

Using the Galatius--Kupers--Randal-Williams framework of cellular $E_2$-algebras, we prove a secondary stability theorem for mapping class groups of nonorientable surfaces. As a corollary, we obtain a new best known stability range for the homology of the mapping class groups of nonorientable surfaces with respect to adding torus holes.