Normal triangulations in o-minimal structures

Abstract

Let 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 φ':|K'|→ |K| of |K| partitioning S₁,...,S_l with φ' definably homotopic to id_|K|. As an application of this result we obtain the semialgebraic Hauptvermutung.