Measured lamination spaces on surfaces and geometric intersection numbers.
type: Item
instance of: scholarly article
title: Measured lamination spaces on surfaces and geometric intersection numbers.
zbMATH Open document ID: 1047.57010
DOI: 10.1016/S0166-8641(03)00220-7
author: Feng Luo
author: Richard A. Stong
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: Let \(\Sigma\) be a compact connected surface. A {curve system} on \(\Sigma\) is a proper essential \(1\)-dimensional submanifold. The space of isotopy classes of curve systems is denoted my \(CS(\Sigma)\). Given two elements \(\alpha,\beta\in CS(\Sigma)\), their {geometric intersection number} will be denoted by \(I(\alpha,\beta)\). Thurston observed that in most cases \(I: CS(\Sigma)\times CS(\Sigma)\to{\mathbb Z} \) behaves like a non-degenerate ``bilinear'' form. Let \(CS_0\subset CS(\Sigma)\) denote the subspace of curve systems whose components are circles. Thurston's space of compactly supported measured laminations on \(\Sigma\), denoted \(ML_{0}(\Sigma)\), is the completion of the pair \((CS_{0}, I)\). The basic result in the theory is that the space \(ML_{0}(\Sigma)\) is homeomorphic to a Euclidean space and that the pairing \(I\) extends to a continuous homogeneous map \(ML_{0}(\Sigma)\times ML_{0}(\Sigma) \to {\mathbb R}\). The aim of the paper under discussion is to give an elementary proof of the basic result for compact surfaces with non-empty boundary. By interpolation, the authors reduce the continuity of the extension of the intersection number to a simple inequality for geometric intersection numbers of arcs inside an octagon.
reviewed by: Michael Heusener
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 2037253
zbMATH Keywords: surfaces
zbMATH Keywords: laminations
zbMATH Keywords: intersection number
zbMATH Keywords: submanifolds
MaRDI profile type: Publication
cites work: Bouts des variétés hyperboliques de dimension 3. (Ends of hyperbolic 3-dimensional manifolds)
cites work: Q2899316
cites work: Q4191989
cites work: Dehn-Thurston coordinates for curves on surfaces
cites work: Tiling the Projective Foliation Space of a Punctured Surface
cites work: Geometric intersection functions and Hamiltonian flows on the space of measured foliations on a surface
cites work: Combinatorics of Train Tracks. (AM-125)
cites work: Q3326476
cites work: On the geometry and dynamics of diffeomorphisms of surfaces
description: scientific article; zbMATH DE number 2037253