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.

     

Weakly quasi-o-minimal models

We introduce the notion of a weakly quasi-o-minimal model and prove that such models lack the independence property. We show that every weakly quasi-o-minimal ordered group is Abelian, every divisible Archimedean weakly quasi-o-minimal ordered group is weakly o-minimal, and every weakly o-minimal quasi-o-minimal ordered group is o-minimal. We also prove that every weakly quasi-o-minimal Archimedean ordered ring with nonzero multiplication is a real closed field that is embeddable into the field of reals.     

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.     

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP₂, though there are also examples for which some or all three of those properties are preserved.     

Weakly o-minimal expansions of ordered fields of finite transcendence degree

Using a work of Diaz concerning algebraic independence of certainsequences of numbers, we prove that if K is a field of finitetranscendence degree over the rationals, then every weakly o-minimalexpansion of (K,,+,·) is polynomially bounded. In thespecial case where K is the field of all real algebraic numbers,we give a proof which makes use of a much weaker result fromtranscendental number theory, namely, the Gelfond–Schneidertheorem. Apart from this we make a couple of observations concerningweakly o-minimal expansions of arbitrary ordered fields of finitetranscendence degree over the rationals. The strongest resultwe prove says that if K is a field of finite transcendence degreeover the rationals, then all weakly o-minimal non-valuationalexpansions of (K,,+,·) are power bounded.     

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.     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

The Field of Reals with Multisummable Series and the Exponential Function

We show that the field of real numbers with multisummable realpower series is model complete, o-minimal and polynomially bounded.Further expansion by the exponential function yields again amodel complete and o-minimal structure which is exponentiallybounded, and in which the Gamma function on the positive realline is definable. 2000 Mathematics Subject Classification:primary 03C10, 32B05, 32B20; secondary, 26E05.     

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.     

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.

     

Isomorphisms of adjoint Chevalley groups over integral domains

It is shown that every automorphism of an adjoint Chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism, a graph automorphism and an inner automorphism while every isomorphism between simple adjoint Chevalley groups can be expressed uniquely as the product of a ring isomorphism, a graph isomorphism and an inner automorphism. The isomorphisms between the elementary subgroups are also found having analogous expressions.

     

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.     

On existence of tame Harrison map

We present here two new criteria for existence of a tame Harrison map of two formally real algebraic function fields over a fixed real closed field of constants. The first criterion (c.f. Theorem 2.5) shows that a square class group isomorphism is a tame Harrison map if it induces an isomorphism of the coproduct rings of residue Witt rings. The other result (c.f. Proposition 3.5) associates a tame Harrison map to an integral quaternion-symbol equivalence.      

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on simple and semisimple Lie groups. We show that the Lie group topology on such a group S is very rigid: every “abstract” isomorphism between S and a locally compact and σ-compact group Γ is automatically a homeomorphism, provided that S is absolutely simple. If S is complex, then noncontinuous field automorphisms of the complex numbers have to be considered, but that is all. We obtain similar results for semisimple groups.     

K-Theory of Surfaces at the Prime 2

We prove that for smooth surfaces over real closed fields, and a class of smooth projective surfaces over a real number field, the map between mod 2 algebraic and étale K-theory is an isomorphism in sufficiently large degrees. For a class of smooth projective surfaces over a real closed field, including rational surfaces, complete intersections and K3-surfaces over the real numbers, we explicate the abutment of the mod 2 motivic cohomology to algebraic K-theory spectral sequence.     

Infinitely Peano differentiable functions in polynomially bounded o-minimal structures

Let R be an o-minimal expansion of a real closed field. We show that the definable infinitely Peano differentiable functions are smooth if and only if R is polynomially bounded.     

Linear Groups Definable in o-Minimal Structures

We study subgroups G of GL(n, R) definable in o-minimal expansions M = (R, +, · ,…) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, power functions, or an exponential function definable in M. (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser–Tits conjecture for real closed fields.     

On orbit types of semialgebraically proper actions

In this paper we consider semialgebraically proper actions of semialgebraic groups on semialgebraic sets. Let G be a semialgebraic group. We prove that every semialgebraically proper G-set has only finitely many orbit types.     

Picard-Vessiot theory for real fields

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard-Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.     

Permutation Groups in o-Minimal Structures

In this paper we develop a structure theory for transitive permutationgroups definable in o-minimal structures. We fix an o-minimalstructure M, a group G definable in M, and a set and a faithfultransitive action of G on definable in M, and talk of the permutationgroup (G, ). Often, we are concerned with definably primitivepermutation groups (G, ); this means that there is no propernon-trivial definable G-invariant equivalence relation on ,so definable primitivity is equivalent to a point stabiliserG being a maximal definable subgroup of G. Of course, sinceany group definable in an o-minimal structure has the descendingchain condition on definable subgroups [23] we expect many questionson definable transitive permutation groups to reduce to questionson definably primitive ones. Recall that a group G definable in an o-minimal structure issaid to be connected if there is no proper definable subgroupof finite index. In some places, if G is a group definable inM we must distinguish between definability in the full ambientstructure M and G-definability, which means definability inthe pure group G:= (G, .); for example, G is G-definably connectedmeans that G does not contain proper subgroups of finite indexwhich are definable in the group structure. By definable, wealways mean definability in M. In some situations, when thereis a field R definable in M, we say a set is R-semialgebraic,meaning that it is definable in (R, +, .). We call a permutationgroup (G, ) R-semialgebraic if G, and the action of G on canall be defined in the pure field structure of a real closedfield R. If R is clear from the context, we also just write‘semialgebraic’.