Group covers, o-minimality, and categoricity

TL;DR

This paper investigates the model theory of group covers in o-minimal structures, focusing on categoricity and differences with algebraic group covers, and explores a topological question related to Milnor's conjecture.

Contribution

It provides new insights into the model-theoretic properties of group covers in o-minimal structures and highlights differences from complex algebraic group covers.

Findings

Categoricity properties differ from complex algebraic cases.

Analysis of finite central extensions and their topological nature.

Discussion of Milnor's conjecture in the context of o-minimal structures.

Abstract

We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case of covers of complex algebraic groups studied by Zilber and his students. We also discuss from a model-theoretic point of view the following question, related to "Milnor's conjecture": is a finite central extension (as an abstract group) of a compact Lie group also a topological extension?