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 such that the quotient G/G, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G 7→ G/G sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group G/G and the o-minimal spectrum G̃ of G. We prove that G/G is a topological quotient of G̃. We thus obtain a natural homomorphism Ψ∗ from the cohomology of G/G to the (Čech-)cohomology of G̃. We show that if G satisfies a suitable contractibility conjecture then G̃ is acyclic in Čech cohomology and Ψ∗ is an isomorphism. Finally we prove the conjecture in some special cases.