Low-dimensional homology groups of mapping class groups: a survey

TL;DR

This survey compiles known results on the first and second homology groups of surface mapping class groups, providing a comprehensive overview of current knowledge in the field.

Contribution

It offers a complete list of known results on low-dimensional homology groups of mapping class groups, summarizing recent progress and open questions.

Findings

First and second homology groups are fully classified.

Some results on higher (co)homology are included.

The survey highlights key open problems in the area.

Abstract

In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.