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A Primer on Mapping Class Groups (PMS-49)$
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Benson Farb and Dan Margalit

Print publication date: 2011

Print ISBN-13: 9780691147949

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691147949.001.0001

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The Symplectic Representation and the Torelli Group

The Symplectic Representation and the Torelli Group

(p.162) Chapter Six The Symplectic Representation and the Torelli Group
A Primer on Mapping Class Groups (PMS-49)

Benson Farb

Dan Margalit

Princeton University Press

This chapter discusses the basic properties and applications of a symplectic representation, denoted by Ψ‎, and its kernel, called the Torelli group. After describing the algebraic intersection number as a symplectic form, the chapter presents three different proofs of the surjectivity of Ψ‎, each illustrating a different theme. It also illustrates the usefulness of the symplectic representation by two applications to understanding the algebraic structure of Mod(S). First, the chapter explains how this representation is used by Serre to prove the theorem that Mod(Sɡ) has a torsion-free subgroup of finite index. It thens uses the symplectic representation to prove, following Ivanov, the following theorem of Grossman: Mod(Sɡ) is residually finite. It also considers some of the pioneering work of Dennis Johnson on the Torelli group. In particular, a Johnson homomorphism is constructed and some of its applications are given.

Keywords:   symplectic representation, algebraic intersection number, Torelli group, Mod(S), finite index, Dennis Johnson, Johnson homomorphism

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