Homological stability of spin mapping class groups and quadratic symplectic groups

TL;DR

This paper investigates the homological stability of spin mapping class groups and quadratic symplectic groups, providing improved results, optimal stability away from prime 2, and detailed descriptions of their first homology groups.

Contribution

It advances understanding of homological stability for these groups using cellular E2-algebras, achieving optimal results and describing their first homology groups.

Findings

Improved stability results for spin mapping class groups and quadratic symplectic groups.

Proved optimal stability for spin mapping class groups away from prime 2.

Provided full descriptions of the first homology groups.

Abstract

We study the homological stability of spin mapping class groups of surfaces and of quadratic symplectic groups using cellular $E_2$-algebras. We get improvements in their stability results, which for the spin mapping class groups we show to be optimal away from the prime $2$. We also prove that in both cases the $\mathbb{F}_2$-homology satisfies secondary homological stability. Finally, we give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups.