Geometry & Topology

An elementary approach to the mapping class group of a surface

Bronisław Wajnryb

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We consider an oriented surface S and a cellular complex X of curves on S, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S, following the ideas of Hatcher–Thurston and Harer.

Article information

Geom. Topol., Volume 3, Number 1 (1999), 405-466.

Received: 7 January 1999
Revised: 18 November 1999
Accepted: 6 December 1999
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F05: Generators, relations, and presentations 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F38: Other groups related to topology or analysis 57M60: Group actions in low dimensions

mapping class group surface curve complex group presentation


Wajnryb, Bronisław. An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), no. 1, 405--466. doi:10.2140/gt.1999.3.405.

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