A crossed homomorphism for groups acting on the circle

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.