Circle action of the punctured mapping class group and cross homomorphism

TL;DR

This paper provides a new geometric interpretation of a specific cohomology generator of the punctured mapping class group using circle actions and rotation numbers, connecting different existing constructions.

Contribution

It introduces a novel geometric perspective on the generator of a cohomology group, linking circle actions with winding and rotation number constructions.

Findings

New geometric interpretation of the cohomology generator

Equivalence of circle action and winding number constructions

Clarification of the generator's geometric nature

Abstract

In the following short note, we give a new geometric interpretation of the generator of the infinite cyclic group $H^1(\text{Mod}(S_{g,1});H^1(S_g;\mathbb{Z}))$ (this computation is proved by Morita). There are several constructions of this class given by Earle, Morita, Trapp and Furuta. The construction we give here uses the action of $\text{Mod}(S_{g,1})$ on the circle and its rotation numbers. We also show that our construction is the same as the construction by Furuta and Trapp using winding numbers.