Homological stability for mapping class groups of surfaces

TL;DR

This paper provides a comprehensive proof of Harer's homological stability theorem for surface mapping class groups, incorporating recent improvements to establish the best known stability range.

Contribution

It offers a complete, detailed proof of Harer's stability theorem with the latest enhancements, consolidating previous advancements.

Findings

Established the most extensive stability range to date.

Unified various improvements into a comprehensive proof.

Confirmed the robustness of homological stability for mapping class groups.

Abstract

We give a complete and detailed proof of Harer's stability theorem for the homology of mapping class groups of surfaces, with the best stability range presently known. This theorem and its proof have seen several improvements since Harer's original proof in the mid-80's, and our purpose here is to assemble these many additions.