Higher homotopy of groups definable in o-minimal structures

TL;DR

This paper demonstrates that for definably compact groups in o-minimal structures, the higher homotopy groups mirror those of their associated compact Lie groups, leading to uniform homotopy types and contractible universal covers.

Contribution

It establishes the isomorphism of higher homotopy groups between definably compact groups and their Lie group counterparts, revealing new homotopy invariants in o-minimal groups.

Findings

Higher homotopy groups of definably compact groups match those of associated Lie groups

All abelian definably compact groups of the same dimension are homotopy equivalent

Universal covers of these groups are contractible

Abstract

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.