PL and differential topology in o-minimal structure

TL;DR

This paper explores the extension of piecewise linear and differential topology concepts within o-minimal structures over real closed fields, establishing foundational results like the o-minimal Hauptvermutung and manifold tameness.

Contribution

It proves the uniqueness of polyhedron structures on compact definable sets and shows that many topological problems over R can be translated into the o-minimal setting.

Findings

Compact definable sets are definably homeomorphic to polyhedra.

Unique PL manifold structures on compact definable manifolds.

Topological problems over R can be transferred to o-minimal structures.

Abstract

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.