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December 2007 O-minimal spectra, infinitesimal subgroups and cohomology
Alessandro Berarducci
J. Symbolic Logic 72(4): 1177-1193 (December 2007). DOI: 10.2178/jsl/1203350779

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” $G^{00}$ 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 exact 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 (Čech-)cohomology of $\widetilde{G}$. We show that if $G^{00}$ satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and $\Psi^*$ is an isomorphism. Finally we prove the conjecture in some special cases.

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Alessandro Berarducci. "O-minimal spectra, infinitesimal subgroups and cohomology." J. Symbolic Logic 72 (4) 1177 - 1193, December 2007. https://doi.org/10.2178/jsl/1203350779

Information

Published: December 2007
First available in Project Euclid: 18 February 2008

zbMATH: 1131.03015
MathSciNet: MR2371198
Digital Object Identifier: 10.2178/jsl/1203350779

Subjects:
Primary: 03C64, 03H05, 22E15

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 4 • December 2007
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