By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.
"Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup." J. Symbolic Logic 74 (3) 891 - 900, September 2009. https://doi.org/10.2178/jsl/1245158089