Mário J. Edmundo

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DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

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 … 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 ℤ n ; 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.

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Structure theorems for o-minimal expansions of groups

2000-03-01

Mário J. Edmundo

Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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The universal covering homomorphism in o‐minimal expansions of groups

2007-09-26

Mário J. Edmundo , Pantelis E. Eleftheriou

Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : … Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : $ \tilde G $ → G is a locally definable covering homomorphism and π 1 ( G ) is isomorphic to the o‐minimal fundamental group π ( G ) of G defined using locally definable covering homomorphisms. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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SHEAF COHOMOLOGY IN o-MINIMAL STRUCTURES

2006-12-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures. Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

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Locally definable groups in o-minimal structures

2005-05-18

Mário J. Edmundo

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Comparison theorems for o-minimal singular (co)homology

2008-04-09

Mário J. Edmundo , Arthur Woerheide

Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories. Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories.

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Covers of groups definable in o-minimal structures

2005-01-01

Mário J. Edmundo

In this paper we develop the theory of covers for locally definable groups in o-minimal structures. In this paper we develop the theory of covers for locally definable groups in o-minimal structures.

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A remark on divisibility of definable groups

2005-09-28

Mário J. Edmundo

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an … We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an o-minimal expansion of a field, k ∈ ℕ and pk : G → G is the definable map given by pk (x ) = xk for all x ∈ G , then we have |(pk )–1(x )| ≥ kr for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Poincaré - Verdier duality in o-minimal structures

2010-01-01

Mário J. Edmundo , Luca Prelli

Here we prove a Poincaré - 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. Here we prove a Poincaré - 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.

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The six Grothendieck operations on o-minimal sheaves

2019-04-01

Mário J. Edmundo , Luca Prelli

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O-minimal cohomology and definably compact definable groups

2000-01-01

Mário J. Edmundo

Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the … Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the Peterzil-Steinhorn problem on the existence of torsion points on N-definably compact N-definable abelian groups. We compute the cohomology rings of N-definably compact N-definable groups, and we prove an o-minimal analog of the Poincare duality theorem, the Alexander dualti theorem, the Lefschetz duality theorem and the Lefschetz fixed point theorem.

Discrete subgroups of locally definable groups

2013-03-22

Alessandro Berarducci , Mário J. Edmundo , Marcello Mamino

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Corrigendum to “Locally definable groups in o-minimal structures” [J. Algebra 301 (2006) 194–223]

2008-08-01

Elías Baro , Mário J. Edmundo

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Integration of positive constructible functions against Euler characteristic and dimension

2006-03-30

Raf Cluckers , Mário J. Edmundo

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Corrigendum to: “Transfer methods for o-minimal topology”

2007-09-01

Alessandro Berarducci , Mário J. Edmundo , Margarita Otero

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Invariance results for definable extensions of groups

2010-07-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

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On Pillay's conjecture in the general case

2017-03-17

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

On definably proper maps

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli

In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map … In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spac

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The universal covering map in o-minimal expansions of groups

2013-06-19

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

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O-MINIMAL CECH COHOMOLOGY

2007-10-18

Mário J. Edmundo , Nicholas J. Peatfield

Journal Article o-MINIMAL ČECH COHOMOLOGY Get access Mário J. Edmundo, Mário J. Edmundo CMAF Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal Search for other works by … Journal Article o-MINIMAL ČECH COHOMOLOGY Get access Mário J. Edmundo, Mário J. Edmundo CMAF Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal Search for other works by this author on: Oxford Academic Google Scholar Nicholas J. Peatfield Nicholas J. Peatfield † Department of Mathematics, University of Bristol, Royal Fort Annex, Bristol BS8 1UJ, UK †Corresponding author. Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 59, Issue 2, June 2008, Pages 213–220, https://doi.org/10.1093/qmath/ham036 Published: 17 October 2007 Article history Revision received: 06 June 2007 Published: 17 October 2007

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Coverings by open cells

2014-01-28

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

Definable group extensions in semi‐bounded o‐minimal structures

2009-11-17

Mário J. Edmundo , Pantelis E. Eleftheriou

Abstract In this note we show: Let R = 〈 R , <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a … Abstract In this note we show: Let R = 〈 R , <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 R m , +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Sheaves on $\mathcal T$-topologies

2016-01-01

Mário J. Edmundo , Luca Prelli

The aim of this paper is to give a unifying description of various constructions of sites (subanalytic, semialgebraic, o-minimal) and consider the corresponding theory of sheaves. The method used applies … The aim of this paper is to give a unifying description of various constructions of sites (subanalytic, semialgebraic, o-minimal) and consider the corresponding theory of sheaves. The method used applies to a more general context and gives new results in semialgebraic and o-minimal sheaf theory.

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The six Grothendieck operations on o-minimal sheaves

2014-01-04

Mário J. Edmundo , Luca Prelli

In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) … In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Kunneth formula; (v) local and global Verdier duality.

Invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi></mml:math>-minimal cohomology with definably compact supports

2015-10-22

Mário J. Edmundo , Luca Prelli

In this paper we find general criteria for invariance and finiteness results for o-minimal cohomology in an arbitrary o-minimal structure. We apply our criteria and obtain new invariance and finiteness … In this paper we find general criteria for invariance and finiteness results for o-minimal cohomology in an arbitrary o-minimal structure. We apply our criteria and obtain new invariance and finiteness results for o-minimal cohomology in o-minimal expansions of ordered groups and for the o-minimal cohomology of definably compact definable groups in arbitrary o-minimal structures.

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Sheaves on T-topologies

2010-02-03

Mário J. Edmundo , Luca Prelli

The aim of this paper is to give a unifying description of various constructions (subanalytic, semialgebraic, o-minimal site) using the notion of T-topology. We then study the category of T-sheaves. The aim of this paper is to give a unifying description of various constructions (subanalytic, semialgebraic, o-minimal site) using the notion of T-topology. We then study the category of T-sheaves.

A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably … We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

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Erratum to: "Covers of groups definable in o-minimal structures" [Illinois J. Math. 49 (2005), no. 1, 99–120

2007-07-01

Mário J. Edmundo

In this short note we point out two errors in our paper "Covers of groups definable in o-minimal structures" [Illinois J. Math.49 (2005), 99-120], and we show how to correct … In this short note we point out two errors in our paper "Covers of groups definable in o-minimal structures" [Illinois J. Math.49 (2005), 99-120], and we show how to correct these errors.

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A fixed point theorem in o-minimal structures

2007-01-01

Mário J. Edmundo

Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps. Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.

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An introduction to o-minimal structures

2000-01-01

Mário J. Edmundo

The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these … The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal structures. We omit here any reference to the pure model theory of o-minimal structures and to the theory of groups and rings definable in o-minimal structures.

Covering definable manifolds by open definable subsets

2007-12-03

Mário J. Edmundo

. Let N be an o-minimal expansion of a real closed field. We show that if X is a Hausdorff definable manifold, then X can be covered by finitely many … . Let N be an o-minimal expansion of a real closed field. We show that if X is a Hausdorff definable manifold, then X can be covered by finitely many open definable subsets which are definably homeomorphic to open balls and the intersection of any two open definable subsets of this covering is a finite union of elements of the covering. We also mention the importance of this result in the solution of the torsion point problem for definably compact definable groups.

The Lefschetz coincidence theorem in o-minimal expansions of fields

2009-07-18

Mário J. Edmundo , Arthur Woerheide

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Fundamental group in o-minimal structures with definable Skolem functions

2021-04-08

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

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On definably proper maps

2014-04-26

Mário J. Edmundo , Marcello Mamino , Luca Prelli

In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if … In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in the category of definable spaces. We give several other characterizations of definably proper including one involving the existence of limits of definable types. We also prove the basic properties of definably proper maps and the invariance of definably proper in elementary extensions and o-minimal expansions.

Fundamental group in o-minimal structures with definable Skolem functions

2018-07-25

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an … In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we 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 allows 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.

The six Grothendieck operations on o-minimal sheaves

2014-04-16

Mário J. Edmundo , Luca Prelli

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Cohomology of algebraic varieties over non-archimedean fields

2020-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology … We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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Invariance of o-minimal cohomology with definably compact supports

2012-01-01

Mário J. Edmundo , Luca Prelli

In this paper we find general criteria to ensure that, in an arbitrary o-minimal structure, the o-minimal cohomology without supports and with definably compact supports of a definable space with … In this paper we find general criteria to ensure that, in an arbitrary o-minimal structure, the o-minimal cohomology without supports and with definably compact supports of a definable space with coefficients in a sheaf is invariant in elementary extensions and in o-minimal expansions. We also prove the o-minimal analogue of Wilder's finiteness theorem in this context.

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Invariance of o-minimal cohomology with definably compact supports

2012-05-28

Mário J. Edmundo , Luca Prelli

On the o-minimal Hilbert's fifth problem

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal … Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$, the $k$-torsion subgroups of $G;$ the o-minimal cohomology algebra over ${\mathbb Q}$ of $G.$ As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

On the Euler characteristic of definable groups

2011-02-01

Mário J. Edmundo

We show that in an arbitrary o-minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if … We show that in an arbitrary o-minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o-minimal Euler characteristic of the quotient is non zero; (ii) every infinite, definably connected, definably compact definable group has a non trivial torsion point (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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On freely generated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>E</mml:mi></mml:math>-subrings

2008-12-19

Mário J. Edmundo , Giuseppina Terzo

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On solvable groups and rings definable in o-minimal structures

2017-03-29

Mário J. Edmundo

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Integration of positive constructible functions against Euler characteristic and dimension

2006-01-01

Raf Cluckers , Mário J. Edmundo

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et integration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411 - 416] on motivic integration, we … Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et integration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411 - 416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula.

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Covers of groups definable in o-minimal structures

2001-01-01

Mário J. Edmundo

We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an … We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an $\N$-definably connected $\N$-definable group $G$ we have $1\to π_1(G)\to \tilde{G}\stackrel{p}\to G\to 1$ in the category of strictly properly $\bigvee $-definable groups with strictly properly $\bigvee $-definable homomorphisms, where $π_1(G)$ is the o-minimal fundamental group of $G$.

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

2010-03-23

Mário J. Edmundo , Luca Prelli

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. 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.

Solvable groups definable in o-minimal structures

2000-01-01

Mário J. Edmundo

Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove … Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we describe the N-definable G-modules and the N-definable rings.

Cohomology of algebraic varieties over non-archimedean fields

2022-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf … Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K . For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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On definable Skolem functions and trichotomy

2022-01-01

Bruno Dinis , Mário J. Edmundo

In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union … In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over $\emptyset $ and finitely many open intervals each a union of a $\emptyset $-definable family of group-intervals with fixed positive elements.

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On Proper Direct Image in o-Minimal Expansions of Groups

2024-07-30

Mário J. Edmundo , Luca Prelli

Abstract We show that the formalism of the six Grothendieck operations holds in the sub-category of definably locally closed definable subsets equipped with the o-minimal site in o-minimal expansions of … Abstract We show that the formalism of the six Grothendieck operations holds in the sub-category of definably locally closed definable subsets equipped with the o-minimal site in o-minimal expansions of ordered groups.

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Cohomology of algebraic varieties over non-archimedean fields

2022-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

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On definable Skolem functions and trichotomy

2022-01-01

Bruno Dinis , Mário J. Edmundo

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Fundamental group in o-minimal structures with definable Skolem functions

2021-04-08

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

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Cohomology of algebraic varieties over non-archimedean fields

2020-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

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The six Grothendieck operations on o-minimal sheaves

2019-04-01

Mário J. Edmundo , Luca Prelli

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Fundamental group in o-minimal structures with definable Skolem functions

2018-07-25

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

Fundamental group in o-minimal structures with definable Skolem functions

2018-01-01

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

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On solvable groups and rings definable in o-minimal structures

2017-03-29

Mário J. Edmundo

On Pillay's conjecture in the general case

2017-03-17

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Sheaves on $\mathcal T$-topologies

2016-01-01

Mário J. Edmundo , Luca Prelli

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Invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi></mml:math>-minimal cohomology with definably compact supports

2015-10-22

Mário J. Edmundo , Luca Prelli

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On the o-minimal Hilbert's fifth problem

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

On definably proper maps

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli

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The Lefschetz coincidence theorem in o-minimal expansions of fields

2015-01-01

Mário J. Edmundo , Arthur Woerheide

In this paper we prove the Lefschetz coincidence theorem in o-minimal expansions of fields using the o-minimal singular homology and cohomology. In this paper we prove the Lefschetz coincidence theorem in o-minimal expansions of fields using the o-minimal singular homology and cohomology.

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On definably proper maps

2014-04-26

Mário J. Edmundo , Marcello Mamino , Luca Prelli

The six Grothendieck operations on o-minimal sheaves

2014-04-16

Mário J. Edmundo , Luca Prelli

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Coverings by open cells

2014-01-28

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

The six Grothendieck operations on o-minimal sheaves

2014-01-04

Mário J. Edmundo , Luca Prelli

The six Grothendieck operations on o-minimal sheaves

2014-01-01

Mário J. Edmundo , Luca Prelli

In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) … In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) K\"unneth formula; (v) local and global Verdier duality.

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On definably proper maps

2014-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli

The universal covering map in o-minimal expansions of groups

2013-06-19

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

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Discrete subgroups of locally definable groups

2013-03-22

Alessandro Berarducci , Mário J. Edmundo , Marcello Mamino

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Coverings by open cells

2013-01-01

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells. We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

Invariance of o-minimal cohomology with definably compact supports

2012-05-28

Mário J. Edmundo , Luca Prelli

Invariance of o-minimal cohomology with definably compact supports

2012-01-01

Mário J. Edmundo , Luca Prelli

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On the Euler characteristic of definable groups

2011-02-01

Mário J. Edmundo

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A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

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Invariance results for definable extensions of groups

2010-07-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

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

2010-03-23

Mário J. Edmundo , Luca Prelli

Sheaves on T-topologies

2010-02-03

Mário J. Edmundo , Luca Prelli

Poincaré - Verdier duality in o-minimal structures

2010-01-01

Mário J. Edmundo , Luca Prelli

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Sheaves on T-topologies

2010-01-01

Mário J. Edmundo , Luca Prelli

Poincaré-Verdier duality in o-minimal structures

2010-01-01

Mário J. Edmundo , Luca Prelli

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Definable group extensions in semi‐bounded o‐minimal structures

2009-11-17

Mário J. Edmundo , Pantelis E. Eleftheriou

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The Lefschetz coincidence theorem in o-minimal expansions of fields

2009-07-18

Mário J. Edmundo , Arthur Woerheide

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On freely generated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>E</mml:mi></mml:math>-subrings

2008-12-19

Mário J. Edmundo , Giuseppina Terzo

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Corrigendum to “Locally definable groups in o-minimal structures” [J. Algebra 301 (2006) 194–223]

2008-08-01

Elías Baro , Mário J. Edmundo

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Comparison theorems for o-minimal singular (co)homology

2008-04-09

Mário J. Edmundo , Arthur Woerheide

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Covering definable manifolds by open definable subsets

2007-12-03

Mário J. Edmundo

O-MINIMAL CECH COHOMOLOGY

2007-10-18

Mário J. Edmundo , Nicholas J. Peatfield

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The universal covering homomorphism in o‐minimal expansions of groups

2007-09-26

Mário J. Edmundo , Pantelis E. Eleftheriou

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Corrigendum to: “Transfer methods for o-minimal topology”

2007-09-01

Alessandro Berarducci , Mário J. Edmundo , Margarita Otero

Erratum to: "Covers of groups definable in o-minimal structures" [Illinois J. Math. 49 (2005), no. 1, 99–120

2007-07-01

Mário J. Edmundo

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A fixed point theorem in o-minimal structures

2007-01-01

Mário J. Edmundo

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SHEAF COHOMOLOGY IN o-MINIMAL STRUCTURES

2006-12-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures. Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

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Integration of positive constructible functions against Euler characteristic and dimension

2006-03-30

Raf Cluckers , Mário J. Edmundo

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Integration of positive constructible functions against Euler characteristic and dimension

2006-01-01

Raf Cluckers , Mário J. Edmundo

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A remark on divisibility of definable groups

2005-09-28

Mário J. Edmundo

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Locally definable groups in o-minimal structures

2005-05-18

Mário J. Edmundo

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Coauthor	Papers Together
Luca Prelli	22
Marcello Mamino	9
Pantelis E. Eleftheriou	5
Giuseppina Terzo	4
Bruno Dinis	4
Arthur Woerheide	3
Nicholas J. Peatfield	3
G. O. Jones	2
Pablo Cubides Kovacsics	2
Margarita Otero	2
Jinhe Ye	2
Alessandro Berarducci	2
Janak Ramakrishnan	2
Raf Cluckers	2
Elías Baro	1

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01

Ya’acov Peterzil , Charles Steinhorn

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

SHEAF COHOMOLOGY IN o-MINIMAL STRUCTURES

2006-12-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures. Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

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DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

Mário J. Edmundo , Margarita Otero

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On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Structure theorems for o-minimal expansions of groups

2000-03-01

Mário J. Edmundo

Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01

Ya’acov Peterzil , Sergei Starchenko

Real Algebraic Geometry

1998-01-01

Frédéric Mangolte , Jean-Philippe Rolin , Kurdyka Krzysztof +2 more

Transfer methods for o-minimal topology

2003-09-01

Alessandro Berarducci , Margarita Otero

Abstract 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 … Abstract 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 φ M to φ R 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.

Groups definable in ordered vector spaces over ordered division rings

2007-12-01

Pantelis E. Eleftheriou , Sergei Starchenko

Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 … Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 G , ⊕, e G 〉 an n -dimensional group definable in M . We show that if G is definably compact and definably connected with respect to the t -topology, then it is definably isomorphic to a ‘definable quotient group’ U/L , for some convex V -definable subgroup U of 〈 M n , +〉 and a lattice L of rank n . As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L .

Poincaré - Verdier duality in o-minimal structures

2010-01-01

Mário J. Edmundo , Luca Prelli

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Homology of Locally Semialgebraic Spaces

1991-01-01

Hans Delfs

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o-MINIMAL FUNDAMENTAL GROUP, HOMOLOGY AND MANIFOLDS

2002-04-01

Alessandro Berarducci , Margarita Otero

The definable fundamental group of a definable set in an o-minimal expansion of a field is computed. This is achieved by proving the relevant case of the o-minimal van Kampen … The definable fundamental group of a definable set in an o-minimal expansion of a field is computed. This is achieved by proving the relevant case of the o-minimal van Kampen theorem. This result is applied to show that if the geometrical realization of a simplicial complex over an o-minimal expansion of a field is a definable manifold of dimension not 4, then its geometrical realization over the reals is a topological manifold.

Definably simple groups in o-minimal structures

2000-02-24

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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Comparison theorems for o-minimal singular (co)homology

2008-04-09

Mário J. Edmundo , Arthur Woerheide

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Linear Groups Definable in o-Minimal Structures

2002-01-01

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Euler characteristic in semialgebraic and other o-minimal groups

1994-10-01

Adam Strzeboński

Returning to semi-bounded sets

2009-06-01

Ya’acov Peterzil

Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an … Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension N such that either N is a reduct of an ordered vector space, or there is an o-minimal structure , with the same universe but of different language from N , with (i) Every definable set in N is definable in , and (ii) has an elementary substructure in which every bounded interval admits a definable real closed field. As a result certain questions about definably compact groups can be reduced to either ordered vector spaces or expansions of real closed fields. Using the known results in these two settings, the number of torsion points in definably compact abelian groups in expansions of ordered groups is given. Pillay's Conjecture for such groups follows.

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

Sheaves on Subanalytic Sites

2008-12-31

Luca Prelli

In [7] the authors introduced the notion of ind-sheaves and defined the six Grothendieck operations in this framework. As a byproduct, they obtained subanalytic sheaves and the six Grothendieck operations … In [7] the authors introduced the notion of ind-sheaves and defined the six Grothendieck operations in this framework. As a byproduct, they obtained subanalytic sheaves and the six Grothendieck operations on them. The aim of this paper is to give a direct construction of the six Grothendieck operations in the framework of subanalytic sites avoiding the heavy machinery of ind-sheaves. As an application, we show how to recover the subanalytic sheaves _O_t and _O_w of temperate and Whitney holomorphic functions respectively.

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The universal covering homomorphism in o‐minimal expansions of groups

2007-09-26

Mário J. Edmundo , Pantelis E. Eleftheriou

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Real algebraic geometry

2014-01-21

Jacek Bochnak , Michel Coste , Marie-Françoise Roy

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- … 1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

1996-01-01

Andrew O.M. Wilkie

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Locally Semialgebraic Spaces

1985-01-01

Hans Delfs , Manfred Knebusch

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Simple algebraic and semialgebraic groups over real closed fields

2000-06-13

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, 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.

Linear O-minimal structures

1993-02-01

James Loveys , Ya’acov Peterzil

Geometric categories and o-minimal structures

1996-08-01

Lou van den Dries , Chris Miller

Sheaves of continuous definable functions

1988-12-01

Anand Pillay

Let M be an o-minimal structure or a p -adically closed field. Let be the space of complete n -types over M equipped with the following topology: The basic open … Let M be an o-minimal structure or a p -adically closed field. Let be the space of complete n -types over M equipped with the following topology: The basic open sets of are of the form Ũ = { p ∈ S n ( M ): U ∈ p } for U an open definable subset of M n . is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K [ X 1 , …, X n ]; see [CR].) We will equip with a sheaf of L M -structures (where L M is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th( M ) has definable Skolem functions, then if p ∈ , it follows that M ( p ), the definable ultrapower of M at p , can be factored through M p , the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ N n and f is an M -definable (partial) function defined at a , then there is an open M -definable set U ⊂ N n with a ∈ U , and a continuous M -definable function g : U → N such that g(a) = f(a) . In the case that M is an o-minimal expansion of a real closed field (or M is a p -adically closed field), it turns out that M(p) can be recovered as the unique quotient of M p which is an elementary extension of M .

A survey on groups definable in o-minimal structures

2008-05-22

Margarita Otero

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 … 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:

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On the homology of algebraic varieties over real closed fields.

1982-09-01

Manfred Knebusch , Hans Delfs

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Covers of groups definable in o-minimal structures

2005-01-01

Mário J. Edmundo

In this paper we develop the theory of covers for locally definable groups in o-minimal structures. In this paper we develop the theory of covers for locally definable groups in o-minimal structures.

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Corrigendum to: “Transfer methods for o-minimal topology”

2007-09-01

Alessandro Berarducci , Mário J. Edmundo , Margarita Otero

Non-archimedean tame topology and stably dominated types

2010-01-01

Ehud Hrushovski , François Loeser

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the … Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it. In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$. When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values. The space $\hat {V}$ is obtained by defining a topology on the pro-definable set of stably dominated types on $V$. The key result is the construction of a pro-definable strong retraction of $\hat {V}$ to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.

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Lectures on Algebraic Topology

1995-01-01

Albrecht Dold

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Invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi></mml:math>-minimal cohomology with definably compact supports

2015-10-22

Mário J. Edmundo , Luca Prelli

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Generic sets in definably compact groups

2007-01-01

Ya’acov Peterzil , Anand Pillay

A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i

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On Pillay's conjecture in the general case

2017-03-17

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Interpretable groups are definable

2014-03-18

Pantelis E. Eleftheriou , Ya’acov Peterzil , Janak Ramakrishnan

We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product … We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.

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A trichotomy theorem for o-minimal structures

1998-11-01

Ya’acov Peterzil , Sergei Starchenko

Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples … Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

A structure theorem for semibounded sets in the reals

1992-09-01

Ya’acov Peterzil

In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λ a › a ∈ℝ , … In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λ a › a ∈ℝ , all the bounded semi-algebraic (that is, -definable) sets are definable. Said differently, every such is an expansion of = ‹ℝ, +, <, λ a , B i › a ∈ℝ, i ∈ I where { B i } i ∈ I is the collection of all bounded semialgebraic sets and the λ a 's are scalar multiplication by a . In [PSS] (see Theorem 1.2 below) it was shown that the structure is a proper reduct of ; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that is the only reduct properly between ℳ and . As a first step towards this result, we investigate in this paper the definable sets in reducts such as . (We point out that ‘definable’ will always mean ‘definable with parameters’.) Definition 1.1. Let X ⊆ ℝ n . X is called semi-bounded if it is definable in the structure ‹ℝ, +, <, λ a , B 1 , …, B k › a ∈ℝ , where the B i 's are bounded subsets of ℝ n . The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the “stretching” of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power. The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in §4.

o-MINIMAL COHOMOLOGY: FINITENESS AND INVARIANCE RESULTS

2009-12-01

Alessandro Berarducci , Antongiulio Fornasiero

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the … The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language.

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p-adic and Real Subanalytic Sets

1988-07-01

Jan Denef , Lou van den Dries

On definably proper maps

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli

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A generalization of the Tarski-Seidenberg theorem, and some nondefinability results

1986-01-01

Lou van den Dries

This article points out some remarkable facts implicit in the results of Lojasiewicz [LI] and Gabrielov [Ga].An important consequence of Tarski's work [T] on the elementary theory of the reals … This article points out some remarkable facts implicit in the results of Lojasiewicz [LI] and Gabrielov [Ga].An important consequence of Tarski's work [T] on the elementary theory of the reals is a characterization of the sets which are elementarily definable from addition and multiplication on R. Allowing arbitrary reals as constants, this characterization consists of the identification of the definable sets with the semialgebraic sets.(A semialgebraic subset of R m is by definition a finite union of sets of the form {x e R w : p(x) = 0, q x (x) > 0,...,q k (x) > 0} where p,q 1 ,...,q k are real polynomials.)The fact that the system of semialgebraic sets is closed under definability is also known as the Tarski-Seidenberg theorem, and this property, together with the topological finiteness phenomena that go with it-triangulability of semialgebraic sets [L2, Gi], generic triviality of semialgebraic maps [Ha]-make the theory of semialgebraic sets a useful analytic-topological tool.Below we extend the system of semialgebraic sets in such a way that the Tarski-Seidenberg property, i.e., closure under definability, and the topological finiteness phenomena are preserved.The polynomial growth property of semialgebraic functions is also preserved.This extended system contains the arctangent function on R, the sine function on any bounded interval, the exponential function e x on any bounded interval, but not the exponential function on all of R. (And it couldn't possibly contain the sine function on all of R without sacrificing the finiteness phenomena, and a lot more.)As a corollary we obtain that neither the exponential function on R, nor the set of integers, is definable from addition, multiplication, and the restrictions of the sine and exponential functions to bounded intervals.Questions of this type have puzzled logicians for a long time.(There still remain, of course, countless unsolved problems of this sort.)In a more positive spirit Tarski [T, p. 45] asked to extend his results so as to include, besides the algebraic operations on R, certain transcendental elementary functions like e x ; the theorem below is a partial answer.(More recently, Hovanskii [Ho, p. 562] and the author [VdDl, VdD2] asked similar questions, and in [VdD3] we

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Sheaves of continuous definable functions

1988-12-01

Anand Pillay

Locally definable homotopy

2009-05-08

Elías Baro , Margarita Otero

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A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007-03-01

Ehud Hrushovski , Ya’acov Peterzil

Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions … Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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Invariance results for definable extensions of groups

2010-07-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

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A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

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