The Structure Of Classical Diffeomorphism Groups
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The Structure Of Classical Diffeomorphism Groups
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Author : Deborah Ajayi
language : en
Publisher:
Release Date : 2014-01-15
The Structure Of Classical Diffeomorphism Groups written by Deborah Ajayi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
The Structure Of Classical Diffeomorphism Groups
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Author : Augustin Banyaga
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
The Structure Of Classical Diffeomorphism Groups written by Augustin Banyaga and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.
The Structure Of Classical Diffeomorphism Groups
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Author : Augustin Banyaga
language : en
Publisher: Springer
Release Date : 1997-03-31
The Structure Of Classical Diffeomorphism Groups written by Augustin Banyaga and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-03-31 with Mathematics categories.
The book introduces and explains most of the main techniques and ideas in the study of the structure of diffeomorphism groups. A quite complete proof of Thurston's theorem on the simplicity of some diffeomorphism groups is given. The method of the proof is generalized to symplectic and volume-preserving diffeomorphisms. The Mather-Thurston theory relating foliations with diffeomorphism groups is outlined. A central role is played by the flux homomorphism. Various cohomology classes connected with the flux are defined on the group of diffeomorphisms. The main results on the structure of diffeomorphism groups are applied to showing that classical structures are determined by their automorphism groups, a contribution to the Erlanger Program of Klein. Audience: Graduate students and researchers in mathematics and physics.
Geometry Topology And Dynamics
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Author : François Lalonde
language : en
Publisher: American Mathematical Soc.
Release Date : 1998
Geometry Topology And Dynamics written by François Lalonde and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Mathematics categories.
This is a collection of papers written by leading experts. They are all clear, comprehensive, and origianl. The volume covers a complete range of exciting and new developments in symplectic and contact geometries.
Infinite Dimensional Lie Groups In Geometry And Representation Theory
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Author : Augustin Banyaga
language : en
Publisher: World Scientific
Release Date : 2002-07-12
Infinite Dimensional Lie Groups In Geometry And Representation Theory written by Augustin Banyaga and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-07-12 with Science categories.
This book constitutes the proceedings of the 2000 Howard conference on “Infinite Dimensional Lie Groups in Geometry and Representation Theory”. It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and established researchers in the field of geometry and its applications to mathematical physics.
The Geometry Of The Group Of Symplectic Diffeomorphism
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Author : Leonid Polterovich
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
The Geometry Of The Group Of Symplectic Diffeomorphism written by Leonid Polterovich and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The group of Hamiltonian diffeomorphisms Ham(M, 0) of a symplectic mani fold (M, 0) plays a fundamental role both in geometry and classical mechanics. For a geometer, at least under some assumptions on the manifold M, this is just the connected component of the identity in the group of all symplectic diffeomorphisms. From the viewpoint of mechanics, Ham(M,O) is the group of all admissible motions. What is the minimal amount of energy required in order to generate a given Hamiltonian diffeomorphism I? An attempt to formalize and answer this natural question has led H. Hofer [HI] (1990) to a remarkable discovery. It turns out that the solution of this variational problem can be interpreted as a geometric quantity, namely as the distance between I and the identity transformation. Moreover this distance is associated to a canonical biinvariant metric on Ham(M, 0). Since Hofer's work this new ge ometry has been intensively studied in the framework of modern symplectic topology. In the present book I will describe some of these developments. Hofer's geometry enables us to study various notions and problems which come from the familiar finite dimensional geometry in the context of the group of Hamiltonian diffeomorphisms. They turn out to be very different from the usual circle of problems considered in symplectic topology and thus extend significantly our vision of the symplectic world.
Structure And Regularity Of Group Actions On One Manifolds
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Author : Sang-hyun Kim
language : en
Publisher: Springer Nature
Release Date : 2021-11-19
Structure And Regularity Of Group Actions On One Manifolds written by Sang-hyun Kim and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-11-19 with Mathematics categories.
This book presents the theory of optimal and critical regularities of groups of diffeomorphisms, from the classical work of Denjoy and Herman, up through recent advances. Beginning with an investigation of regularity phenomena for single diffeomorphisms, the book goes on to describes a circle of ideas surrounding Filipkiewicz's Theorem, which recovers the smooth structure of a manifold from its full diffeomorphism group. Topics covered include the simplicity of homeomorphism groups, differentiability of continuous Lie group actions, smooth conjugation of diffeomorphism groups, and the reconstruction of spaces from group actions. Various classical and modern tools are developed for controlling the dynamics of general finitely generated group actions on one-dimensional manifolds, subject to regularity bounds, including material on Thompson's group F, nilpotent groups, right-angled Artin groups, chain groups, finitely generated groups with prescribed critical regularities, and applications to foliation theory and the study of mapping class groups. The book will be of interest to researchers in geometric group theory.
Groups Of Circle Diffeomorphisms
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Author : Andrés Navas
language : en
Publisher: University of Chicago Press
Release Date : 2011-06-01
Groups Of Circle Diffeomorphisms written by Andrés Navas and has been published by University of Chicago Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-01 with Mathematics categories.
In recent years scholars from a variety of branches of mathematics have made several significant developments in the theory of group actions. Groups of Circle Diffeomorphisms systematically explores group actions on the simplest closed manifold, the circle. As the group of circle diffeomorphisms is an important subject in modern mathematics, this book will be of interest to those doing research in group theory, dynamical systems, low dimensional geometry and topology, and foliation theory. The book is mostly self-contained and also includes numerous complementary exercises, making it an excellent textbook for undergraduate and graduate students.
A Brief Introduction To Symplectic And Contact Manifolds
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Author : Augustin Banyaga
language : en
Publisher: World Scientific
Release Date : 2016-08-08
A Brief Introduction To Symplectic And Contact Manifolds written by Augustin Banyaga and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-08 with Mathematics categories.
The book introduces the basic notions in Symplectic and Contact Geometry at the level of the second year graduate student. It also contains many exercises, some of which are solved only in the last chapter.We begin with the linear theory, then give the definition of symplectic manifolds and some basic examples, review advanced calculus, discuss Hamiltonian systems, tour rapidly group and the basics of contact geometry, and solve problems in chapter 8. The material just described can be used as a one semester course on Symplectic and Contact Geometry.The book contains also more advanced material, suitable to advanced graduate students and researchers.
Diffeomorphisms Of Elliptic 3 Manifolds
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Author : Sungbok Hong
language : en
Publisher: Springer
Release Date : 2012-08-29
Diffeomorphisms Of Elliptic 3 Manifolds written by Sungbok Hong and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-08-29 with Mathematics categories.
This work concerns the diffeomorphism groups of 3-manifolds, in particular of elliptic 3-manifolds. These are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, now known to be exactly the closed 3-manifolds that have a finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to its diffeomorphism group is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. The main results establish the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens spaces L(m,q) with m at least 3. Additional results imply that for a Haken Seifert-fibered 3 manifold V, the space of Seifert fiberings has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background