Simple Loops on Surfaces and Their Intersection Numbers

1998/01/06 by Feng Luo, Luo, Feng
Mathematics · #57 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57

paper · pdf · doi:10.48550/arxiv.math/9801018

42 pages, 29 figures

arxiv created 1998/01/06 · arxiv updated 2009/11/30

Abstract

Given a compact orientable surface Σ, let \Cal S(Σ) be the set of isotopy classes of essential simple loops on Σ. We determine a complete set of relations for a function from \Cal S(Σ) to \bold Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in \bold R^{\Cal S(Σ)} and P(\bold R^{\Cal S(Σ)}) defining Thurston's space of measured laminations and Thurston's compactification of the Teichmüller space. These equations are not only piecewise integral linear but also semi-real algebraic.

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• Geometric intersections of loops on surfaces
• Lengths of simple loops on surfaces with hyperbolic metrics
• Loops in surfaces and star-fillings
• Minimal intersection of curves on surfaces
• Cauchy Inequality and the Space of Measured Laminations