The Torelli groups for genus 2 and 3 surfaces

Abstract

THE purpose of this paper is to show that the genus 2 Torelli group is free on infinitely many Dehn twists on separating curves. Moreover, the set of free generators can be identified with the set of splittings of the homology of a genus 2 surface into two subspaces mutually orthogonal and unimodular with respect to the intersection pairing. In addition, it is shown that the third integer homology of the genus 3 Torelli group naturally contains an infinitely generated free abelian group. This is a permutation module for the symplectic modular group. The method used is a study of the period mapping to Siegel space. In Sections 2 and 3 we review some background material. Section 4 shows that the genus 2 Torelli group is free. Section 5 contains additional background material and the result on the third homology. Section 6 gives a homological application.