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.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

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.     

Bifurcation sets of definable functions in o-minimal structures

In this work we answer a question stated by Loi and Zaharia concerning trivialization of definable functions off the bifurcation set: we prove that definable functions are trivial off the bifurcation set, and the trivialization can be chosen definable.

     

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.     

Some definable properties of sets in non-valuational weakly o-minimal structures

Let \({\mathcal {M}}=(M,<,+,\cdot ,\ldots )\) be a non-valuational weakly o-minimal expansion of a real closed field \((M,<,+,\cdot )\). In this paper, we prove that \({\mathcal {M}}\) has a \(C^r\)-strong cell decomposition property, for each positive integer r, a best analogous result from Tanaka and Kawakami (Far East J Math Sci (FJMS) 25(3):417–431, 2007). We also show that curve selection property holds in non-valuational weakly o-minimal expansions of ordered groups. Finally, we extend the notion of definable compactness suitable for weakly o-minimal structures which was examined for definable sets (Peterzil and Steinhorn in J Lond Math Soc 295:769–786, 1999), and prove that a definable set is definably compact if and only if it is closed and bounded.     

Fiberwise properties of definable sets and functions in o-minimal structures

Summary We examine how in any o-minimal expansion of a dense linear order, fiberwise open implies pecewise open for sets definable with parameters, and fiberwise continuous implies piecewise continuous for functions definable with parameters.     

Tame properties of sets and functions definable in weakly o-minimal structures

Let ${{\mathcal{M}}=(M, <, \ldots )}$ be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in ${{\mathcal{M}}}$ which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in ${{\mathcal{M}}}$ satisfy an extended version of IVP. After introducing a weak version of definable connectedness in ${{\mathcal{M}}}$ , we prove that strong cells in ${{\mathcal{M}}}$ are weakly definably connected, so every set definable in ${{\mathcal{M}}}$ is a finite union of its weakly definably connected components, provided that ${{\mathcal{M}}}$ has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in ${{\mathcal{M}}}$ and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure ${{\mathcal{M}}}$ having cell decomposition, satisfies NDG, i.e. every definable function in ${{\mathcal{M}}}$ has no dense graph.     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

A trichotomy theorem for o-minimal structures

Let M = M, <, ... be alinearly ordered structure. We defineM to be o-minimal if every definable subset of M is a finiteunion of intervals. Classical examples are ordered divisibleabelian groups and real closed fields. We prove a trichotomytheorem for the structure that an arbitraryo-minimal M can induceon a neighbourhood of any a in M. Roughly said, one of the followingholds:

(i) a is trivial (technical term), or

(ii) a has aconvex neighbourhood on which M induces the structureof anordered vector space, or

(iii) a is contained in an open intervalon which M inducesthe structure of an expansion of a real closedfield.

The proof uses ‘geometric calculus’ whichallows one to recover a differentiable structure by purely geometricmethods. 1991 Mathematics Subject Classification: primary 03C45;secondary 03C52, 12J15, 14P10.     

Strongly minimal expansions of (C, +) definable in o-minimal fields

We characterize those functions f: definable in o-minimalexpansions of the reals for which the structure (,+, f) is stronglyminimal: such functions must be complex constructible, possiblyafter conjugating by a real matrix. In particular we prove aspecial case of the Zilber Dichotomy: an algebraically closedfield is definable in certain strongly minimal structures whichare definable in an o-minimal field.     

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.     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

     

Definable one dimensional structures in o-minimal theories

Let N be a structure definable in an o-minimal structure M and p ∈ S _N (N), a complete N-1-type. If dim _M (p) = 1, then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N-definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N-definable real closed field.     

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.     

Expansions of algebraically closed fields in o-minimal structures

We develop a notion of differentiability over an algebraically closed field K of characteristic zero with respect to a maximal real closed subfield R. We work in the context of an o-minimal expansion ? \cal {R} of the field R and obtain many of the standard results in complex analysis in this setting. In doing so we use the topological approach to complex analysis developed by Whyburn and others. We then prove a model theoretic theorem that states that the field R is definable in every proper expansion of the field K all of whose atomic relations are definable in ? \cal {R} . One corollary of this result is the classical theorem of Chow on projective analytic sets.