On the homotopy type of definable groups in an o-minimal structure

TL;DR

This paper investigates the homotopy type of definably compact groups in o-minimal structures, establishing a connection with compact Lie groups and analyzing their fundamental groups.

Contribution

It extends the understanding of the homotopy type of definably compact groups by relating their universal covers and fundamental groups to those of associated Lie groups.

Findings

F(G) determines the definable homotopy type of G

Fundamental groups of open subsets in F(G) relate to those in G

The o-minimal fundamental groupoid of G is crucial for analysis

Abstract

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact group. We also show that F(G) determines the definable homotopy type of G. A crucial step is to show that the fundamental group of an open subset of F(G) is isomorphic to the definable fundamental group of its preimage in G. Our results depend on the study of the o-minimal fundamental groupoid of G.