Simple Loops on Surfaces and Their Intersection Numbers

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.