Fundamental group in o-minimal structures with definable Skolem functions

TL;DR

This paper explores the properties of the o-minimal fundamental group within structures that have definable Skolem functions, establishing key topological properties and covering space theory analogous to classical algebraic topology.

Contribution

It proves that definably connected, locally definable manifolds are path connected and have covers by simply connected sets, extending fundamental group theory to o-minimal structures with Skolem functions.

Findings

Definably connected manifolds are uniformly definably path connected.

Existence of universal locally definable covering maps.

Classification of locally definable covering maps and related theorems.

Abstract

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves - from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert - van Kampen theorems.