Simple Loops on Surfaces and Their Intersection Numbers

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