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1. Morse complexes and multiplicative structures

Author

Francois Laudenbach, Hossein Abbaspour, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Université de Nantes - Faculté des Sciences et des Techniques, and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Transversality, High Energy Physics::Lattice, General Mathematics, Boundary (topology), 02 engineering and technology, Morse code, 01 natural sciences, law.invention, Mathematics - Geometric Topology, law, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Condensed Matter::Quantum Gases, 010102 general mathematics, Multiplicative function, Geometric Topology (math.GT), 021001 nanoscience & nanotechnology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], 0210 nano-technology, Manifold (fluid mechanics)
Abstract

In this article we lay out the details of Fukaya's A ∞-structure of the Morse com-plexe of a manifold possibily with boundary. We show that this A ∞-structure is homotopically independent of the made choices. We emphasize the transversality arguments that some fiber product constructions make valid.

Published
2021

3. W. P. Thurston and French mathematics

Author

Athanase Papadopoulos, Francois Laudenbach, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Chebyshev Laboratory, St Petersburg State University (SPbU), Université de Nantes - Faculté des Sciences et des Techniques, and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Low-dimensional topology, History and Overview (math.HO), General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], low-dimensional topology, 01 natural sciences, 01A70, 01A60, 01A61, 57M50, 57R17, 57R30, Mathematics - Geometric Topology, contact structures, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, 57R30 Keywords: William P Thurston: Geometric structures, 0101 mathematics, foliations, Mathematics, Final version, Mathematical society, Mathematics - History and Overview, 010102 general mathematics, Geometric Topology (math.GT), Hyperbolic structures, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Haefliger structures, William P. Thurston: Geometric structures, history of French mathematics Part 1, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], AMS classification: 01A70, [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], history of French mathematics, Hyperbolic struc- tures, Mathematical economics
Abstract

We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appear in the Surveys of the European Mathematical Society.

Published
2019

4. Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds

Author

Olga V. Pochinka, Vyacheslav Zigmuntovich Grines, Francois Laudenbach, Christian Bonatti, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow] (HSE), Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), RFBR (project nos. 15-01-03687-a, 16-51-10005-Ko a), RSF (project no 14-41-00044), BRP at the HSE (project 90) in 2017, ERC Geodycon, ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), National Research University Higher School of Economics [Moscow], Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), European Project : 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON ( 2012 ), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and National Research University Higher School of Economics [Moscow] (HSE)

Subjects
[ MATH.MATH-GT ] Mathematics [math]/Geometric Topology [math.GT], Pure mathematics, Mathematics::Dynamical Systems, General Mathematics, Morse code, 01 natural sciences, law.invention, Equivalence class (music), Mathematics - Geometric Topology, law, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, Heteroclinic lamination, 0101 mathematics, Mathematics::Symplectic Geometry, Characteristic space, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Topological classification, Heteroclinic cycle, Geometric Topology (math.GT), Mathematics::Geometric Topology, Nonlinear Sciences::Chaotic Dynamics, Embedding, 010307 mathematical physics, Diffeomorphism, Morse-Smale diffeomorphism, MSC: 37C05, 37C15, 37C29, 37D15, Topological conjugacy
Abstract

We show that, up to topological conjugation, the equivalence class of a Morse–Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

Published
2019

5. René Thom and an anticipated h-principle

Author

Francois Laudenbach, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Simplex, Immersions, jiggling, Philosophy, Homotopy, 010102 general mathematics, h-principle, 16. Peace & justice, 01 natural sciences, Haefliger structures, MSC : 57R17, 57R30, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, Immersion (mathematics), 010307 mathematical physics, 0101 mathematics, Mathematical economics, foliations
Abstract

International audience; The first part of this article intends to present the role played by Thom in diffusing Smale’s ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale’s idea in what is now known as the h-principle. Here, the h stands for homotopy.Shortly after the astonishing discovery by Smale, Thom gave a conference in Lille (1959) announcing a theorem which would deserve to be named a homological h-principle. The aim of our second part is to comment about this theorem which was completely ignored by the topologists in Paris, but not in Leningrad. We explain Thom’s statement and answer the question whether it is true. The first idea is combinatorial. A beautiful subdivision of the standard simplex emerges from Thom’s article. We connect it with the jiggling technique introduced by W. Thurston in his seminal work on foliations.

Published
2019

6. Haefliger structures and symplectic/contact structures

Author

Francois Laudenbach and Gaël Meigniez

Subjects
Algebra, General Mathematics, Immersion (mathematics), Codimension, Mathematical proof, Mathematics::Geometric Topology, Mathematics::Symplectic Geometry, Manifold, Geometric data analysis, Parametric statistics, Symplectic geometry, Mathematics
Abstract

For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametric versions. The proofs borrow ideas from W. Thurston, like jiggling and inflation. Actually, we are using a more primitive jiggling due to R. Thom.

Published
2016

8. A proof of Reidemeister-Singer’s theorem by Cerf’s methods

Author

Francois Laudenbach

Subjects
Path (topology), Discrete mathematics, Lemma (mathematics), General Medicine, Morse code, Mathematics::Geometric Topology, law.invention, Maxima and minima, law, Simple (abstract algebra), Cerf theory, Mathematics::Symplectic Geometry, Heegaard splitting, Mathematics, Morse theory
Abstract

Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M>2. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.

Published
2014

9. Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

Author

V. Z. Grines, Francois Laudenbach, and Olga V. Pochinka

Subjects
Pure mathematics, Mathematics::Dynamical Systems, 010102 general mathematics, Mathematical analysis, Dimension (graph theory), Function (mathematics), Type (model theory), Morse code, 01 natural sciences, 010305 fluids & plasmas, law.invention, Set (abstract data type), Mathematics (miscellaneous), law, 0103 physical sciences, Attractor, Embedding, Diffeomorphism, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics
Abstract

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors’ previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.

Published
2012

10. Morse 2-jet space and h-principle

Author

Alain Chenciner, Francois Laudenbach, Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Morse singularity, Jet (mathematics), General Mathematics, Morse code, Space (mathematics), 58A20, 58E05, 57R45, 01 natural sciences, law.invention, Section (fiber bundle), Mathematics - Geometric Topology, 2-jet, law, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Cerf theory, Circle-valued Morse theory, Morse theory, Mathematics, Mathematical physics, Holonomic, 010102 general mathematics, Mathematical analysis, h-principle, 2000 MSC: 58A20,58E05 57R45, Geometric Topology (math.GT), 010307 mathematical physics
Abstract

International audience; A section in the 2-jet space of Morse functions is not always homotopic to a holonomic section. We give a necessary condition for being the case and we discuss the sufficiency.

Published
2009

11. Quasi-energy function for diffeomorphisms with wild separatrices

Author

Olga V. Pochinka, Francois Laudenbach, and V. Z. Grines

Subjects
Lyapunov function, Class (set theory), Pure mathematics, Mathematics::Dynamical Systems, General Mathematics, 010102 general mathematics, Mathematical analysis, Cobordism, Function (mathematics), 01 natural sciences, 010101 applied mathematics, symbols.namesake, symbols, Diffeomorphism, 0101 mathematics, Mathematics::Symplectic Geometry, Heegaard splitting, Saddle, Mathematics, Morse theory
Abstract

According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the least number of critical points and construct a quasi-energy function for any diffeomorphism from some class of Morse-Smale diffeomorphisms on the 3-sphere.

Published
2009

12. Self-indexing Energy Function for Morse–Smale Diffeomorphisms on 3-Manifolds

Author

Francois Laudenbach, Olga V. Pochinka, and V. Z. Grines

Subjects
Pure mathematics, Mathematics::Dynamical Systems, General Mathematics, Function (mathematics), Type (model theory), Morse code, Mathematics::Geometric Topology, Manifold, law.invention, law, Saddle point, Diffeomorphism, Mathematics::Symplectic Geometry, Heegaard splitting, Energy (signal processing), Mathematics
Abstract

The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.

Published
2009

15. The energy function for gradient-like diffeomorphisms on 3-manifolds

Author

V. Z. Grines, Olga V. Pochinka, and Francois Laudenbach

Subjects
Lyapunov function, symbols.namesake, Pure mathematics, General Mathematics, Invariant manifold, symbols, Periodic point, Function (mathematics), Energy (signal processing), Morse theory, Mathematics
Published
2008

16. Thurston's Work on Surfaces

Author

Albert Fathi, François Laudenbach, Valentin Poénaru, Albert Fathi, François Laudenbach, and Valentin Poénaru

Subjects
Homeomorphisms, Surfaces, Dynamics, MATHEMATICS / Advanced, MATHEMATICS / Topology
Abstract

This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo-Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory.Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.

Published
2012

22. A Note on the Chas-Sullivan product

Author

Francois Laudenbach, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Pure mathematics, Space (mathematics), Free loop space, 01 natural sciences, Mathematics::Algebraic Topology, law.invention, Mathematics - Geometric Topology, law, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, Chas-Sullivan product, Free loop, 0101 mathematics, Mathematics, intersection product, 010102 general mathematics, Geometric Topology (math.GT), 2000 MSC. 55N10, 57R19, 57R45, 16. Peace & justice, Mathematics::Geometric Topology, 55N10, 57R19, 57R45, Product (mathematics), 010307 mathematical physics, Manifold (fluid mechanics)
Abstract

International audience; We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.

Published
2011

23. Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3

Author

Francois Laudenbach, Gaël Meigniez, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), and Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Pure mathematics, General Mathematics, 010102 general mathematics, open book, Geometric Topology (math.GT), Codimension, Homology (mathematics), 16. Peace & justice, 01 natural sciences, Mathematics::Geometric Topology, Foliations, 010104 statistics & probability, Mathematics - Geometric Topology, MSC[2000]: 57R30, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], Foliation (geology), FOS: Mathematics, 0101 mathematics, Equivalence (formal languages), Haefliger's Gamma-structures, Linear embedding, Mathematics::Symplectic Geometry, Mathematics
Abstract

International audience; We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurston's 1976 regularization theorem without using Mather's homology equivalence.

Published
2011
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24. On the flux of pseudo-Anosov homeomorphisms

Author

Francois Laudenbach, Vincent Colin, Ko Honda, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Department of Mathematics, University of Southern California (USC), and ANR-06-BLAN-0030,SYMPLEXE,Interactions complexe/symplectique : géométrie et dynamique.(2006)

Subjects
Surface (mathematics), Pure mathematics, Mathematics::Dynamical Systems, Mathematics::General Topology, Flux, 01 natural sciences, Mathematics - Geometric Topology, 53C15, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0502 economics and business, FOS: Mathematics, 0101 mathematics, Mathematics, 010102 general mathematics, 05 social sciences, Zero (complex analysis), contact homology, Geometric Topology (math.GT), Mathematics::Geometric Topology, 57M50, Homeomorphism, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], flux, Reeb vector field, pseudo-Anosov, Mathematics - Symplectic Geometry, Open book decomposition, open book decomposition, Symplectic Geometry (math.SG), Geometry and Topology, 050203 business & management, 57M50, 53C15
Abstract

We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero, Comment: 11 pages

Published
2008

25. Essential curves in handlebodies and topological contractions

Author

V. Z. Grines, Francois Laudenbach, department of mathematics, N. Novgorod State University, N. Novgorod State University, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
010102 general mathematics, Geometric Topology (math.GT), compression disk, Topology, 01 natural sciences, 37D15, Mathematics::Geometric Topology, MSC: 57M25, 37D15, 57M25, 37D15, Heegaard splitting, Mathematics - Geometric Topology, North-South diffeomorphism, Compact space, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 57M25, FOS: Mathematics, Geometry and Topology, 0101 mathematics, Handlebody, Mathematics
Abstract

International audience; If $X$ is a compact set, a {\it topological contraction} is a self-embedding $f$ such that the intersection of the successive images $f^k(X)$, $k>0$, consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus $\geq 2$ whose image is essential. Our proof is based on an easy criterion for a simple curve to be essential in a handlebody.

Published
2008

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