Margarita Otero

Definably compact abelian groups

Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤn; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ)n, and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.

A descending chain condition for groups definable in o-minimal structures

We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G 00 of bounded index and G/G 00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

Model Theory with Applications to Algebra and Analysis: A survey on groups definable in o-minimal structures

Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay’s theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following: (1) Interpretability, motivated by an o-minimal version of Cherlin’s conjecture on groups of finite Morley rank (see Sections 4 and 3). (2) The study of the Euler characteristic and the torsion, motivated by a question of Y.Peterzil and C.Steinhorn and results of A.Strzebonski (see Sections 6 and 5). (3) Pillay’s conjectures (see Sections 8 and 7). On interpretability, we have a clear view, with final results in Theorems 4.1 and 4.3 below. Lines of research (2) and (3) can be seen as a way of comparing definable groups with real Lie groups (see Section 2). The best results on the Euler characteristic are those of Theorems 6.3 and 6.5. The study of the torsion begins the study of the algebraic properties of definable groups, and the best result about the algebraic structure of the torsion

Intersection theory for o-minimal manifolds

Abstract We develop an intersection theory for definable C p -manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable C p -homotopies ( p>2 ). In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable C p -homotopies. A. Pillay has shown that any definable group admits an abstract manifold structure. We apply the intersection theory to definable groups after proving an embedding theorem for abstract definably compact C p -manifolds. In particular using the Lefschetz fixed point theorem we show that the Lefschetz number of the identity map on a definably compact group, which in the classical case coincides with the Euler characteristic, is zero.

Locally definable homotopy

Abstract In [E. Baro, M. Otero, On o-minimal homotopy, Quart. J. Math. (2009) 15pp, in press ( doi:10.1093/qmath/hap011 )] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in ⋁ -definable groups — which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.

A Recursive Nonstandard Model of Normal Open Induction

Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peanos axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.

G-linear sets and torsion points in definably compact groups

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G\X) < dim G for some definable

The joint embedding property in normal open induction

{X \subseteq G}