A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}

TL;DR

This paper derives a combinatorial formula for Earle's twisted 1-cocycle on the mapping class group of a surface, relating it to Morita's cocycle and computing it on a hyperelliptic involution.

Contribution

It provides a new explicit combinatorial formula for Earle's cocycle, connecting it with Morita's cocycle and topologically analyzing a hyperelliptic involution.

Findings

Derived a combinatorial formula for Earle's cocycle.

Connected Earle's cocycle with Morita's cocycle.

Computed cocycles on a hyperelliptic involution.

Abstract

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.