Johnson homomorphisms

Abstract

Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S. Morita, are tools for understanding Torelli groups. This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman–Turaev Lie bialgebra, and the work of various authors on cohomological methods for determining the stable image of generalized Johnson homomorphisms. Various open problems and conjectures are included.

Even though the Johnson homomorphisms were originally defined and studied by topologists, they are important in understanding arithmetic properties of mapping class groups and moduli spaces of curves. We define arithmetic Johnson homomorphisms, which extend the generalized Johnson homomorphisms, and explain how the Turaev cobracket constrains their images.