Stratified Polyhedra

Stratified Polyhedra

This is developed along lines now somewhat traditional amongst bundle-theorists, first pioneered by Thom in the smooth category.

Author: David A. Stone

Publisher: Springer

ISBN: 9783540371120

Category: Mathematics

Page: 200

View: 525

Categories: Mathematics

Proceedings on Infinite Dimensional Holomorphy

Proceedings on Infinite Dimensional Holomorphy

252 : D. A. Stone , Stratified Polyhedra . IX , 193 pages . 1972 . DM 18 ,Vol . 253 : V. Komkov , Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems . V , 240 pages . 1972. DM 20 ,Vol .

Author: T.L. Hayden

Publisher: Springer

ISBN: 3540066195

Category: Mathematics

Page: 216

View: 479

Categories: Mathematics

Algebraic and Geometric Topology

Algebraic and Geometric Topology

$1. Stratified polyhedra. A filtered polyhedron (X) = (Xnon-1, . . . .30) is a polyhedron X together with a family of closed subpolyhedra X = Xn PXa-1 ~ .

Author: Kenneth C. Millett

Publisher: Springer

ISBN: 9783540357582

Category: Mathematics

Page: 242

View: 489

Categories: Mathematics

Integer Points in Polyhedra Geometry Number Theory Algebra Optimization

Integer Points in Polyhedra     Geometry  Number Theory  Algebra  Optimization

Let M be a A-lattice polyhedral manifold with boundary 6M, ... By an Eulerian stratification of X we mean a finite collection of disjoint subspaces Xa, ...

Author: Alexander Barvinok

Publisher: American Mathematical Soc.

ISBN: 9780821834596

Category: Convex sets

Page: 191

View: 491

The AMS-IMS-SIAM Summer Research Conference on Integer Points in Polyhedra took place in Snowbird (UT). This proceedings volume contains original research and survey articles stemming from that event. Topics covered include commutative algebra, optimization, discrete geometry, statistics, representation theory, and symplectic geometry. The book is suitable for researchers and graduate students interested in combinatorial aspects of the above fields.
Categories: Convex sets

Monoidal Categories and Topological Field Theory

Monoidal Categories and Topological Field Theory

Stratified. 2-polyhedra. We define (finite) graphs and introduce our key geometric tool, that of a stratified 2-polyhedron.

Author: Vladimir Turaev

Publisher: Birkhäuser

ISBN: 9783319498348

Category: Mathematics

Page: 523

View: 783

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.
Categories: Mathematics

Singularity Theory

Singularity Theory

A stratified vector field on X is a family £ = {Cx}xes of vector fields, ... This problem was solved by Clint McCrory for stratified polyhedra [23, ...

Author: Denis Cheniot

Publisher: World Scientific

ISBN: 9789812707499

Category: Mathematics

Page: 1065

View: 802

The Singularity School and Conference took place in Luminy, Marseille, from January 24th to February 25th 2005. More than 180 mathematicians from over 30 countries converged to discuss recent developments in singularity theory. The volume contains the elementary and advanced courses conducted by singularities specialists during the conference, general lectures on singularity theory, and lectures on applications of the theory to various domains. The subjects range from geometry and topology of singularities, through real and complex singularities, to applications of singularities.
Categories: Mathematics

Singularity Theory

Singularity Theory

A stratified vector field on A is a family Q = {x}xex of vector fields, ... This problem was solved by Clint McCrory for stratified polyhedra (23, ...

Author:

Publisher:

ISBN: 9789814476393

Category:

Page:

View: 789

Categories:

The Topological Classification of Stratified Spaces

The Topological Classification of Stratified Spaces

A filtered space X is a PL stratified space if all the X , are polyhedra and for any two points x , y in a component of some pure stratum there is a PL ...

Author: Shmuel Weinberger

Publisher: University of Chicago Press

ISBN: 0226885666

Category: Mathematics

Page: 283

View: 951

This book provides the theory for stratified spaces, along with important examples and applications, that is analogous to the surgery theory for manifolds. In the first expository account of this field, Weinberger provides topologists with a new way of looking at the classification theory of singular spaces with his original results. Divided into three parts, the book begins with an overview of modern high-dimensional manifold theory. Rather than including complete proofs of all theorems, Weinberger demonstrates key constructions, gives convenient formulations, and shows the usefulness of the technology. Part II offers the parallel theory for stratified spaces. Here, the topological category is most completely developed using the methods of "controlled topology." Many examples illustrating the topological invariance and noninvariance of obstructions and characteristic classes are provided. Applications for embeddings and immersions of manifolds, for the geometry of group actions, for algebraic varieties, and for rigidity theorems are found in Part III. This volume will be of interest to topologists, as well as mathematicians in other fields such as differential geometry, operator theory, and algebraic geometry.
Categories: Mathematics

Extending Intersection Homology Type Invariants to Non Witt Spaces

Extending Intersection Homology Type Invariants to Non Witt Spaces

D. A. Stone, Stratified polyhedra, Lecture Notes in Math., no. 252, Springer Verlag, New York, 1972. Andrei Verona, Stratified mappings – structure and ...

Author: Markus Banagl

Publisher: American Mathematical Soc.

ISBN: 9780821829882

Category: Mathematics

Page: 83

View: 151

Intersection homology theory provides a way to obtain generalized Poincare duality, as well as a signature and characteristic classes, for singular spaces. For this to work, one has had to assume however that the space satisfies the so-called Witt condition. We extend this approach to constructing invariants to spaces more general than Witt spaces. We present an algebraic framework for extending generalized Poincare duality and intersection homology to singular spaces $X$ not necessarily Witt. The initial step in this program is to define the category $SD(X)$ of complexes of sheaves suitable for studying intersection homology type invariants on non-Witt spaces. The objects in this category can be shown to be the closest possible self-dual 'approximation' to intersection homology sheaves.It is therefore desirable to understand the structure of such self-dual sheaves and to isolate the minimal data necessary to construct them. As the main tool in this analysis we introduce the notion of a Lagrangian structure (related to the familiar notion of Lagrangian submodules for $(-1)^k$-Hermitian forms, as in surgery theory). We demonstrate that every complex in $SD(X)$ has naturally associated Lagrangian structures and conversely, that Lagrangian structures serve as the natural building blocks for objects in $SD(X).Our main result asserts that there is in fact an equivalence of categories between $SD(X)$ and a twisted product of categories of Lagrangian structures. This may be viewed as a Postnikov system for $SD(X)$ whose fibers are categories of Lagrangian structures. The question arises as to which varieties possess Lagrangian structures. To begin to answer that, we define the model-class of varieties with an ordered resolution and use block bundles to describe the geometry of such spaces. Our main result concerning these is that they have associated preferred Lagrangian structures, and hence self-dual generalized intersection homology sheaves.
Categories: Mathematics

Introduction to the H principle

Introduction to the H principle

Stratified sets and polyhedra A closed subset S of a manifold V is called stratified if it is presented as N⋃ a union 0 Sj of locally closed submanifolds ...

Author: Y. Eliashberg

Publisher: American Mathematical Soc.

ISBN: 9780821832271

Category: Mathematics

Page: 206

View: 562

In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the fifties that the solvability of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $h$-principle. Two famous examples of the $h$-principle, the Nash-Kuiper $C^1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the $h$-principle. The authors cover two main methods for proving the $h$-principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry. Gromov's famous book ``Partial Differential Relations'', which is devoted to the same subject, is an encyclopedia of the $h$-principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists and analysts will also find much value in this very readable exposition of an important and remarkable topic.
Categories: Mathematics

Surveys on Surgery Theory AM 149 Volume 2

Surveys on Surgery Theory  AM 149   Volume 2

... polyhedra and the topological (CO) versions It is important to realize that Siebenmann didn't just take the topological version of Mather's stratified ...

Author: Sylvain Cappell

Publisher: Princeton University Press

ISBN: 9781400865215

Category: Mathematics

Page: 380

View: 479

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four. In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.
Categories: Mathematics

Surveys on surgery theory papers dedicated to C T C Wall

Surveys on surgery theory   papers dedicated to C T C  Wall

... polyhedra and the topological ( Co ) versions It is important to realize that Siebenmann didn't just take the topological version of Mather's stratified ...

Author: Sylvain Cappell

Publisher: Princeton University Press

ISBN: 0691088144

Category:

Page: 436

View: 456

Categories:

Surveys on Surgery Theory

Surveys on Surgery Theory

... polyhedra and the topological (C") versions It is important to realize that Siebenmann didn't just take the topological version of Mather's stratified ...

Author: Sylvain Cappell

Publisher: Princeton University Press

ISBN: 9780691088150

Category: Mathematics

Page: 380

View: 424

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four. In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.
Categories: Mathematics

C Bundles and Compact Transformation Groups

C   Bundles and Compact Transformation Groups

[32] D. A. Stone, Stratified Polyhedra, Springer-Verlag (Lecture Notes No. 252), Berlin-Heidelberg-N.Y., 1972. [33] R. Thom, Ensembles et morphismes ...

Author: Bruce D. Evans

Publisher: American Mathematical Soc.

ISBN: 9780821822692

Category: Mathematics

Page: 63

View: 627

Categories: Mathematics

Differential Geometry and Topology

Differential Geometry and Topology

The only moment, when this stratification is not generic, is when the manifold passes the ... Each generic 2-dimensional polyhedron is naturally stratified.

Author: R Caddeo

Publisher: World Scientific

ISBN: 9789814553087

Category:

Page: 276

View: 167

This volume contains the courses and lectures given during the workshop on Differential Geometry and Topology held at Alghero, Italy, in June 1992. The main goal of this meeting was to offer an introduction in attractive areas of current research and to discuss some recent important achievements in both the fields. This is reflected in the present book which contains some introductory texts together with more specialized contributions. The topics covered in this volume include circle and sphere packings, 3-manifolds invariants and combinatorial presentations of manifolds, soliton theory and its applications in differential geometry, G-manifolds of low cohomogeneity, exotic differentiable structures on R4, conformal deformation of Riemannian manifolds and Riemannian geometry of algebraic manifolds. Contents:Asystatic G-Manifolds (A Alekseevsky & D Alekseevsky)Les Paquets de Cercles (M Berger)Smooth Structures on Euclidean Spaces (S Demichelis)Surface Theory, Harmonic Maps and Commuting Hamiltonian Flows (D Ferus)Metric Invariants of Kähler Manifolds (M Gromov)On the Sphere Packing Problem and the Proof of Kepler's Conjecture (W Y Hsiang)A 3-Gem Approach to Turaev-Viro Invariants (S L S Lins)Cohomology Operations and Modular Invariant Theory (L Lomonaco)Scalar Curvature and Conformal Deformation of Riemannian Manifolds (A Ratto)Lectures on Combinatorial Presentations of Manifolds (O Viro) Readership: Mathematicians. keywords:
Categories:

Introduction to Piecewise Linear Topology

Introduction to Piecewise Linear Topology

Extension to polyhedra: L.9 Stone, D.: A counterexample in block bundle theory. Topology 9, 11–12 (1970). L.10 Stone, D.: Stratified polyhedra.

Author: Colin P. Rourke

Publisher: Springer Science & Business Media

ISBN: 9783642817359

Category: Mathematics

Page: 126

View: 458

The first five chapters of this book form an introductory course in piece wise-linear topology in which no assumptions are made other than basic topological notions. This course would be suitable as a second course in topology with a geometric flavour, to follow a first course in point-set topology, andi)erhaps to be given as a final year undergraduate course. The whole book gives an account of handle theory in a piecewise linear setting and could be the basis of a first year postgraduate lecture or reading course. Some results from algebraic topology are needed for handle theory and these are collected in an appendix. In a second appen dix are listed the properties of Whitehead torsion which are used in the s-cobordism theorem. These appendices should enable a reader with only basic knowledge to complete the book. The book is also intended to form an introduction to modern geo metric topology as a research subject, a bibliography of research papers being included. We have omitted acknowledgements and references from the main text and have collected these in a set of "historical notes" to be found after the appendices.
Categories: Mathematics

Ends of Complexes

Ends of Complexes

A.M.S. 302, 297-317 (1987) [159] D. Stone, Stratified polyhedra, Lecture Notes in Mathematics 252, Springer (1972) [160] L. Taylor, Surgery on paracompact ...

Author: Bruce Hughes

Publisher: Cambridge University Press

ISBN: 9780521576253

Category: Mathematics

Page: 353

View: 686

A systematic exposition of the theory and practice of ends of manifolds and CW complexes, not previously available.
Categories: Mathematics

Groups of Automorphisms of Manifolds

Groups of Automorphisms of Manifolds

Cerf, J., Le stratification naturelle des espaces des fonctions différentielles réeles et ... Stone, David, Stratified Polyhedra, Lecture Notes in Math, no.

Author: D. Burghelea

Publisher: Springer

ISBN: 9783540375234

Category: Mathematics

Page: 162

View: 475

Categories: Mathematics

Geometry of Subanalytic and Semialgebraic Sets

Geometry of Subanalytic and Semialgebraic Sets

... key of the proofs of III.1.1 and III.1.2, which show Hauptvermutung for a pair of strongly isomorphically stratified polyhedra under certain conditions.

Author: Masahiro Shiota

Publisher: Springer Science & Business Media

ISBN: 9781461220084

Category: Mathematics

Page: 434

View: 798

Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of -or the connected components of-a constructible set (Le. , a locally finite union of differ ences of real analytic sets) need not be constructible (e. g. , R - {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 -=I- O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic.
Categories: Mathematics