PL and differential topology in o-minimal structure

Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable set in R^n is definably homeomorphic to a polyhedron (see [v]). We show uniqueness of the polyhedron up to PL homeomorphisms (o-minimal Hauptvermutung). Hence a compact definable topological manifold admits uniquely a PL manifold structure and is, so to say, tame. We also see that many problems on PL and differential topology over R can be translated to those over the real field.