On definably proper maps

In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in the category of definable spaces. We give several other characterizations of definably proper including one involving the existence of limits of definable types. We also prove the basic properties of definably proper maps and the invariance of definably proper in elementary extensions and o-minimal expansions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1401.084

Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of G. Given a locally definable connected group G (not necessarily definably generated), we prove that the n-torsion subgroup of G is finite and that every zero-dimensional compatible subgroup of G has finite rank. Under a convexity hypothesis we show that every zero-dimensional compatible subgroup of G is finitely generated.Comment: Final version. 17 pages. To appear in Selecta Mathematic

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