A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an o-minimal expansion of a field, k ∈ ℕ and p_k: G → G is the definable map given by p_k(x) = x^k for all x ∈ G, then we have |(p_k)^{-1}(x)| ≥ k^r for all x ∈ G, where r > 0 is the maximal dimension of abelian definable subgroups of G

Locally definable groups in o-minimal structures

In this paper we develop the theory of locally definable groups in o-minimal structures generalizing in this way the theory of definable groups.EPSRC (England) grant GR/M66332; FCT (Portugal) grant SFRH/BPD/6015/200

Covers of groups definable in o-minimal structures

In this paper we develop the theory of covers for locally definable groups in o-minimal structure

Structure theorems for o-minimal expansions of groups

Let R be an o-minimal expansion of an ordered group (R,0,1,+,<) with distinguished positive element 1. We first prove that the following are equivalent: (1) R is semi-bounded, (2) R has no poles, (3) R cannot define a real closed field with domain R and order <, (4) R is eventually linear and (5) every R-definable set is a finite union of cones. As a corollary we get that Th(R) has quantifier elimination and universal axiomatization in the language with symbols for the ordered group operations, bounded R-definable sets and a symbol for each definable endomorphism of the group (R,0,+)

On freely generated E-subrings

In this paper we prove, without assuming Schanuel's conjecture, that the E-subring generated by a real number not definable without parameters in the real exponential field is freely generated. We also obtain a similar result for the complex exponential field.FCT (Funda ção para a Ciência e Tecnologia), program POCTI 2010 (Portugal/FEDER-EU

A note on generic subsets of definable groups

We generalize the theory of generic subsets of definably compact de-finable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.Fundação para a Ciência e a Tecnologia, Financiamento Base 2008 - ISFL/1/20

Comparison theorems for o-minimal singular (co)homology

Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theoriesFCT grant SFRH/BPD/6015/2001; European Research and Training Network HPRN-CT-2001-00271 RAAG

The Lefschetz coincidence theorem in o-minimal expansions of fields

In this paper we prove the Lefschetz coincidence theorem in o-minimal expansions of fields using the o-minimal singular homology and cohomologyFCT (Funda ção para a Ciência e Tecnologia) program POCTI (Portugal/FEDER-EU)

Definable group extensions in semi-bounded o-minimal structures

In this note we show: Let ℛ = 〈 R, <, +, 0,...〉 be a semi-bounded (respectively, linear) o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 Rm, +〉.FCT Financiamento Base 2008 - USFL/1/209; FCT grant SFRH/BPD/35000/200