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2016 The extended Goldman bracket determines intersection numbers for surfaces and orbifolds
Moira Chas, Siddhartha Gadgil
Algebr. Geom. Topol. 16(5): 2813-2838 (2016). DOI: 10.2140/agt.2016.16.2813


In the mid eighties Goldman proved that an embedded closed curve could be isotoped to not intersect a given closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldman’s. The arguments are purely topological, or based on elementary ideas from hyperbolic geometry.

These results are intended to be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three-manifolds. The recognition is based on the structure of the string topology bracket of three-manifolds.


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Moira Chas. Siddhartha Gadgil. "The extended Goldman bracket determines intersection numbers for surfaces and orbifolds." Algebr. Geom. Topol. 16 (5) 2813 - 2838, 2016.


Received: 31 March 2015; Revised: 7 January 2016; Accepted: 20 January 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06653759
MathSciNet: MR3572349
Digital Object Identifier: 10.2140/agt.2016.16.2813

Primary: 57M50

Keywords: Hyperbolic , intersection number , orbifold , surfaces

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 5 • 2016
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