Journal of Symbolic Logic

O-minimal spectra, infinitesimal subgroups and cohomology

Alessandro Berarducci

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Symbolic Logic, Volume 72, Issue 4 (2007), 1177-1193.

First available in Project Euclid: 18 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C64: Model theory of ordered structures; o-minimality 22E15: General properties and structure of real Lie groups 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05]

o-minimality definable group type-definable group Lie group


Berarducci, Alessandro. O-minimal spectra, infinitesimal subgroups and cohomology. J. Symbolic Logic 72 (2007), no. 4, 1177--1193. doi:10.2178/jsl/1203350779.

Export citation