On the o-minimal Hilbert's fifth problem

TL;DR

This paper computes fundamental groups, torsion subgroups, and cohomology of definably compact abelian groups in o-minimal structures, providing a new proof of Pillay's conjecture linking these groups to compact Lie groups.

Contribution

It introduces new methods to analyze the topological invariants of definably compact groups in o-minimal structures and offers a uniform proof of Pillay's conjecture.

Findings

Computed the o-minimal fundamental group of G

Determined the k-torsion subgroups of G

Established the o-minimal cohomology algebra of G

Abstract

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$, the $k$-torsion subgroups of $G;$ the o-minimal cohomology algebra over ${\mathbb Q}$ of $G.$ As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.