Poincar\'e-Verdier duality in o-minimal structures

Here we prove a Poincar\'e-Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.Comment: 23 pages, uses xy-pi

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)

On o-minimal homotopy

Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 20-05-0

Groups definable in o-minimal and NIP settings

In 2004 Pillay conjectured that if G is a definable compact group in a sufficiently saturated o-minimal structure then: 1) G admits a minimal type-definable normal subgroup of bounded index, call it G00. 2) G/G00, equipped with the logic topology, is a compact real Lie group of the same dimension as G. The beauty of this statement is that it offers a surprising connection between the pure lattice of definable sets in definable groups in o-minimal structures and real Lie groups. However, when a proof of the full conjecture was finally found, it was somewhat unsatisfactory from a model theoretic point of view, as it made use of several external tools coming from algebraic topology and other subjects. Nonetheless, it helped to introduce the use of measures in model theory, and made clear that the right framework in which to study these objects was the more general setting of NIP theories. These intuitions recently developed in a new powerful theory of groups and measures in the NIP settings. The aim of this work is therefore to give a concise and self-contained exposition of the theory of groups definable in o-minimal structures as seen from the point of view of NIP theories. A great deal of attention has been paid in using the recent advancements in the subject to eliminate any unnecessary prerequisite from the original proofs, and to rewrite them using the most recent model theoretic viewpoint. Among the other results, we describe a relatively short proof of the compact domination property for definably compact groups

Transfer Methods for O-minimal Topology

Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φM to φℝ and vice versa. Then, we apply these transfer results to give a new proof of a result of M. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field