Let ℛ be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S₁,..., Sl 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,...,Sl with φ' definably homotopic to id|K|. As an application of this result we obtain the semialgebraic Hauptvermutung.
"Normal triangulations in o-minimal structures." J. Symbolic Logic 75 (1) 275 - 288, March 2010. https://doi.org/10.2178/jsl/1264433921