on the o-minimal LS-category

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture.     

Fundamental group in o-minimal structures with definable Skolem functions

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves – from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert–van Kampen theorems.     

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition.     

Locally polynomially bounded structures

We prove a theorem which provides a method for constructingpoints on varieties defined by certain smooth functions. Werequire that the functions be definable in a definably completeexpansion of a real closed field and be locally definable ina fixed o-minimal and polynomially bounded reduct. As an applicationwe show that in certain o-minimal structures, definable functionsare piecewise implicitly defined over the basic functions inthe language.     

Weakly o-minimal structures and real closed fields

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

     

Decomposition into special submanifolds

We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.     

A Convenient Category of Locally Preordered Spaces

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of “locally preordered” spaces. We show that our new category is Cartesian closed that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.     

Invariance results for definable extensions of groups

We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.     

Vanishing homology

In this paper, we introduce a new homology theory devoted to the study of families such as semialgebraic or subanalytic families, and in general, to any family definable in an o-minimal structure (such as Denjoy–Carleman definable, or ln-exp definable sets). The idea is to study the cycles that are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated.     

Definable Compactness and Definable Subgroups of o-Minimal Groups

The paper introduces the notion of definable compactness andwithin the context of o-minimal structures proves several topologicalproperties of definably compact spaces. In particular a definableset in an o-minimal structure is definably compact (with respectto the subspace topology) if and only if it is closed and bounded.Definable compactness is then applied to the study of groupsand rings in o-minimal structures. The main result proved isthat any infinite definable group in an o-minimal structurethat is not definably compact contains a definable torsion-freesubgroup of dimension 1. With this theorem, a complete characterizationis given of all rings without zero divisors that are definablein o-minimal structures. The paper concludes with several examplesillustrating some limitations on extending the theorem.     

Open core and small groups in dense pairs of topological structures

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate.For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Locally injective maps in O-minimal structures without poles are surjective

If is continuous and locally injective, then is in fact surjective and a homeomorphism, provided is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; `without poles' means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.

     

Postnikov factorizations at infinity

We have developed Postnikov sections for Brown–Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces, which is an extension of the proper category. The homotopy fibre of a fibration in the factorization associated with Brown–Grossman groups is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Steenrod homotopy groups. For a space which is first countable at infinity, one of these groups is given by the inverse limit of the homotopy groups of the neighbourhoods at infinity, the other group is isomorphic to the first derived of the inverse limit of this system of groups. In the factorization associated with Steenrod groups the homotopy fibre is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Brown–Grossman homotopy groups. We also obtain a mix factorization containing both kinds of previous factorizations and having homotopy fibres which are Eilenberg–Mac Lane exterior spaces for both kinds of groups.Given a compact metric space embedded in the Hilbert cube, its open neighbourhoods provide the Hilbert cube the structure of an exterior space and the homotopy fibres of the factorizations above are Eilenberg–Mac Lane exterior spaces with respect to inward (or approaching) Quigley groups.     

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.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

Computing o-minimal topological invariants using differential topology

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

     

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

     

Higher homotopy of groups definable in o-minimal structures

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal covers are contractible.     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.