The Torelli group and congruence subgroups of the mapping class group

TL;DR

This paper provides an overview of the Torelli group, Johnson homomorphism, and the second cohomology of level p congruence subgroups of the mapping class group, based on lecture notes from a summer school.

Contribution

It offers a summarized exposition of the Torelli group, Johnson homomorphism, and cohomology of congruence subgroups, integrating previous research and lecture material.

Findings

Description of the Torelli group and Johnson homomorphism

Analysis of the second cohomology group of level p congruence subgroups

Connections to moduli space of curves with level structures

Abstract

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers "The second rational homology group of the moduli space of curves with level structures" and "The Picard group of the moduli space of curves with level structures".