O-minimal spectra, infinitesimal subgroups and cohomology

By recent work on some conjectures of Pillay, each definably compact group $G$ in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' $G^{00}$ such that the quotient $G/G^{00}$, equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor $G\mapsto G/G^{00}$ sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group $G/G^{00}$ and the o-minimal spectrum $\widetilde G$ of $G$. We prove that $G/G^{00}$ is a topological quotient of $\widetilde G$. We thus obtain a natural homomorphism $\Psi^*$ from the cohomology of $G/G^{00}$ to the (\v{C}ech-)cohomology of $\widetilde G$. We show that if $G^{00}$ satisfies a suitable contractibility conjecture then $\widetilde {G^{00}}$ is acyclic in \v{C}ech cohomology and $\Psi^*$ is an isomorphism. Finally we prove the conjecture in some special cases.