Measured lamination spaces on surfaces and geometric intersection numbers. (Q1421994)

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.