Abstract

Let $\scr{R}$ be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S₁,...,S l of its realization $|K|$ in R we prove that there exist a subdivision K' of K and a definable triangulation $\phi ^{\prime}\colon |K^{\prime}|\rightarrow |K|$ of $|K|$ partitioning S₁,...,S l with $\phi ^{\prime}$ definably homotopic to $id_{|K|}$ . As an application of this result we obtain the semialgebraic Hauptvermutung