Mapping class groups and function spaces: a survey

TL;DR

This survey explores the connections between mapping class groups, configuration spaces, and function spaces, focusing on cohomology calculations for low-genus surfaces to understand their algebraic and topological properties.

Contribution

It compiles and reviews existing calculations of cohomology groups for mapping class groups of low-genus surfaces, highlighting their relationships with configuration and function spaces.

Findings

Cohomology groups for low-genus orientable surfaces are systematically calculated.

Cohomology groups for non-orientable surfaces are also analyzed.

The survey consolidates key results linking mapping class groups with configuration and function spaces.

Abstract

This paper is a survey of the relationship between labelled configuration spaces, mapping class groups with marked points and function spaces. In particular, we collect calculations of the cohomology groups for the mapping class groups of low-genus orientable and non-orientable surfaces.