A class function on the mapping class group of an orientable surface and the Meyer cocycle

TL;DR

This paper introduces a new class function on the mapping class group of a surface with boundary, linking it to the signature difference of associated surface bundles, thus providing a novel perspective on the group's cohomology.

Contribution

It defines a $ extbf{QP}^1$-valued class function on the mapping class group and relates it to the Meyer cocycle via surface bundle signatures.

Findings

The class function cobounds a specific 2-cocycle on the mapping class group.

It connects the algebraic structure of the group with geometric invariants.

Provides a new tool for studying the cohomology of surface mapping class groups.

Abstract

In this paper we define a $\mathbf{QP}^1$-valued class function on the mapping class group $\mathcal{M}_{g,2}$ of a surface $\Sigma_{g,2}$ of genus $g$ with two boundary components. Let $E$ be a $\Sigma_{g,2}$ bundle over a pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we obtain a closed surface bundle over $P$. We have another closed surface bundle by gluing to $E$ the product of $P$ and two disks. The sign of our class function cobounds the 2-cocycle on $\mathcal{M}_{g,2}$ defined by the difference of the signature of these two surface bundles over $P$.