Compact and finite-type support in the homology of big mapping class groups

TL;DR

This paper investigates whether the homology of big mapping class groups contains non-trivial classes supported on compact or finite-type subsurfaces, providing comprehensive results for surfaces with positive genus.

Contribution

It offers an almost-complete characterization for surfaces with positive genus and a partial characterization for genus-zero surfaces regarding homology classes supported on subsurfaces.

Findings

Homology classes supported on compact subsurfaces are characterized for positive genus surfaces.

Homological stability techniques are applied to understand the structure of these classes.

Partial results are obtained for genus-zero surfaces.

Abstract

For any infinite-type surface $S$, a natural question is whether the homology of its mapping class group contains any non-trivial classes that are supported on (i) a compact subsurface or (ii) a finite-type subsurface. Our purpose here is to study this question, in particular giving an almost-complete answer when the genus of $S$ is positive (including infinite) and a partial answer when the genus of $S$ is zero. Our methods involve the notion of shiftable subsurfaces as well as homological stability for mapping class groups of finite-type surfaces.