O-minimal spectra, infinitesimal subgroups and cohomology

Original · EN

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'' G⁰⁰ such that the quotient G/G⁰⁰, equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor G G/G⁰⁰ sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group G/G⁰⁰ and the o-minimal spectrum G of G. We prove that G/G⁰⁰ is a topological quotient of G. We thus obtain a natural homomorphism Ψ* from the cohomology of G/G⁰⁰ to the (Čech-)cohomology of G. We show that if G⁰⁰ satisfies a suitable contractibility conjecture then G⁰⁰ is acyclic in Čech cohomology and Ψ* is an isomorphism. Finally we prove the conjecture in some special cases.