Fe b 20 10 PL AND DIFFERENTIAL TOPOLOGY IN O-MINIMAL STRUCTURE Masahiro SHIOTA

Arguments on PL(=piecewise linear) topology work over any ordered field in the same way as over R, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,·). 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 R.