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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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Mapping Class Group Basics

Mapping Class Group Basics

(p.44) Chapter Two Mapping Class Group Basics
A Primer on Mapping Class Groups (PMS-49)

Benson Farb

Dan Margalit

Princeton University Press

This chapter explains the basics of the mapping class group of a surface. It first provides the definition and examples before computing the mapping class group in essentially all of the cases where it can be computed directly. This includes the case of the disk, the annulus, the torus, and the pair of pants. An important method, the Alexander method, emerges as a tool for such computations and is used to prove whether a homeomorphism is or is not homotopically trivial, and whether two homeomorphisms are homotopic or not. One of the computations performed using the Alexander method is a classical fundamental theorem of Dehn.

Keywords:   mapping class group, surface, disk, annulus, torus, Alexander method, homeomorphism

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