  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.

Logic