Marked fatgraph complexes and surface automorphisms

TL;DR

This paper extends combinatorial methods in surface automorphism theory to subgroups defined by a homomorphism to an arbitrary group, providing explicit cocycles for Johnson homomorphisms and the Earle class.

Contribution

It introduces a combinatorial framework for subgroups of the mapping class group relative to a homomorphism, generalizing classical cases and explicitly representing key cohomology classes.

Findings

Explicit cocycle for the first Johnson homomorphism with target Λ^3 K

Representation of the Earle class by an explicit cocycle

Extension of combinatorial surface automorphism theory to new subgroups

Abstract

Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $\Lambda ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.