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O-minimal spectra, infinitesimal subgroups and cohomology

Published online by Cambridge University Press:  12 March 2014

Alessandro Berarducci
Alessandro Berarducci*
Affiliation:
Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy. E-mail: [email protected] URL: www.dm.unipi.it/~berardu
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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 of G. We prove that G/G00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G00 to the (Čech-)cohomology of . We show that if G00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.

Keywords

o-minimalitydefinable grouptype-definable groupLie group
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1177 - 1193
Copyright
Copyright © Association for Symbolic Logic 2007

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