O-Minimal Spectra, Infinitesimal Subgroups and Cohomology

Journal of Symbolic Logic 72 (4):1177 - 1193 (2007)

Abstract
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 exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum G̃ of G. We prove that G/G00 is a topological quotient of G̃. We thus obtain a natural homomorphism ψ* from the cohomology of G/G00 to the (Čech-)cohomology of G̃. We show that if G00 satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and ψ* is an isomorphism. Finally we prove the conjecture in some special cases
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DOI 10.2178/jsl/1203350779
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References found in this work BETA

Type-Definability, Compact Lie Groups, and o-Minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.
Definably Compact Abelian Groups.Mário J. Edmundo & Margarita Otero - 2004 - Journal of Mathematical Logic 4 (02):163-180.
Intersection Theory for o-Minimal Manifolds.Alessandro Berarducci & Margarita Otero - 2001 - Annals of Pure and Applied Logic 107 (1-3):87-119.
A Remark on Divisibility of Definable Groups.Mário J. Edmundo - 2005 - Mathematical Logic Quarterly 51 (6):639-641.

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Citations of this work BETA

Connected Components of Definable Groups, and o-Minimality II.Annalisa Conversano & Anand Pillay - 2015 - Annals of Pure and Applied Logic 166 (7-8):836-849.

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