## Journal of Symbolic Logic

### O-minimal spectra, infinitesimal subgroups and cohomology

Alessandro Berarducci

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 18 February 2008

https://projecteuclid.org/euclid.jsl/1203350779

Digital Object Identifier
doi:10.2178/jsl/1203350779

Mathematical Reviews number (MathSciNet)
MR2371198

Zentralblatt MATH identifier
1131.03015

#### Citation

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