Lagrangian mapping class groups from a group homological point of view

TL;DR

This paper investigates Lagrangian subgroups of the mapping class group of a surface using group (co)homology, revealing their abelianizations, bounds on second homology, and insights into higher degree (co)homology, including the second homology of genus 3 surfaces.

Contribution

It provides the first detailed group (co)homological analysis of Lagrangian mapping class groups and determines the second homology of the genus 3 surface mapping class group.

Findings

Determined the abelianizations of Lagrangian mapping class groups.

Established lower bounds for their second homology.

Computed the second homology of the genus 3 surface mapping class group.

Abstract

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.