A crossed homomorphism for groups acting on the circle

TL;DR

This paper constructs a crossed homomorphism for groups acting on the circle using the Poincaré translation number, relating it to the Euler class, and applies it to the mapping class group of a punctured sphere.

Contribution

It introduces a new crossed homomorphism for circle actions and connects it to the Euler class via spectral sequences, answering a question by Calegari and Chen.

Findings

Constructed a crossed homomorphism using circle actions and translation numbers.

Related the homomorphism to the Euler class through Hochschild--Serre spectral sequence.

Provided an explicit form of the crossed homomorphism for the mapping class group of a punctured sphere.

Abstract

We construct a crossed homomorphism by using a group action on the circle and the Poincar\'{e} translation number. We relate it to the Euler class of the action in terms of the Hochschild--Serre spectral sequence. As an application, we answer a question of Calegari and Chen, which is on an explicit form of a certain crossed homomorphism on the mapping class group of the sphere minus a Cantor set.