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 -> G/G(00) sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G(00) and the o-minimal spectrum (G) over tilde of G. We prove that G/G(00) is a topological quotient of (G) over tilde. We thus obtain a natural homomorphism psi* from the cohomology of G/G(00) to the (Cech-)cohomology of (G) over tilde. We show that if G(00) satisfies a suitable contractibility conjecture then (G(00)) over tilde is acyclic in Cech cohomology and psi* is an isomorphism. Finally we prove the conjecture in some special cases.