On the homology of big mapping class groups

TL;DR

This paper proves the acyclicity of the mapping class group of the one-holed Cantor tree surface and determines the homology of related infinite-type surfaces, answering a recent open question.

Contribution

It establishes the homology of big mapping class groups for a class of infinite-type surfaces, extending previous results and introducing new methods.

Findings

Mapping class group of the one-holed Cantor tree surface is acyclic

Homology of the once-punctured Cantor tree surface is determined

Results apply to a broad class of binary tree surfaces

Abstract

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces.