دسته‌بندی لوسترنیک-شنیرلمن (نظرسنجی‌ها و مونوگراف‌ها ریاضی)

"Lusternik-Schnirelmann category is like a Picasso painting. Looking at category from different perspectives produces completely different impressions of category's beauty and applicability." Lusternik-Schnirelmann category is a subject with ties to both algebraic topology and dynamical systems. The authors take LS-category as the central theme, and then develop topics in topology and dynamics around it. Included are exercises and many examples. The book presents the material in a rich, expository style. The book provides a unified approach to LS-category, including foundational material on homotopy theoretic aspects, the Lusternik-Schnirelmann theorem on critical points, and more advanced topics such as Hopf invariants, the construction of functions with few critical points, connections with symplectic geometry, the complexity of algorithms, and category of 3-manifolds. This is the first book to synthesize these topics. It takes readers from the very basics of the subject to the state of the art. Prerequisites are few: two semesters of algebraic topology and, perhaps, differential topology. It is suitable for graduate students and researchers interested in algebraic topology and dynamical systems. Readership: Graduate students and research mathematicians interested in algebraic topology and dynamical systems. Lusternik-schnirelmann Category Is A Subject With Ties To Both Algebraic Topology And Dynamical Systems. This Book Provides A Unified Approach To Ls-category, Including Foundational Material On Homotopy Theoretic Aspects Of The Subject, The Lusternik-schnirelmann Theorem On Critical Points And More Advanced Topics Such As Hopf Invariants, The Construction Of Functions With Few Critical Points, Connections With Symplectic Geometry, The Complexity Of Algorithms And Category Of 3-manifolds. This Is The First Book Which Takes Ls-category As Its Central Theme And Develops Topics In Topology And Dynamics Around It. As Such, It Leads From The Very Basics Of The Subject To The Present-day State Of The Art. The Prerequisites For Reading The Book Are Few: Two Semesters Of Algebraic Topology And, Perhaps, Differential Topology.--book Jacket. 1.3. The Lusternik-schnirelmann Theorem 7 -- 1.4. Sums, Homotopy Invariance And Mapping Cones 13 -- 1.5. Products And Fibrations 17 -- 1.6. The Whitehead And Ganea Formulations Of Category 22 -- 1.7. Axioms And Category 33 -- Chapter 2. Lower Bounds For Ls-category 47 -- 2.2. Ganea Fibrations Of A Product 49 -- 2.3. Toomer's Invariant 52 -- 2.4. Weak Category 55 -- 2.5. Conilpotency Of A Suspension 57 -- 2.6. Suspension Of The Category 60 -- 2.7. Category Weight 62 -- 2.8. Comparison Theorem 66 -- Chapter 3. Upper Bounds For Category 75 -- 3.2. First Properties Of Upper Bounds 76 -- 3.3. Geometric Category Is Not A Homotopy Invariant 79 -- 3.4. Strong Category And Category Differ By At Most One 82 -- 3.5. Cone-length 83 -- 3.6. Stabilization Of Ball Category 92 -- 3.7. Constraints Implying Equality Of Category And Upper Bounds 98 -- Chapter 4. Localization And Category 105 -- 4.2. Localization Of Groups And Spaces 106 -- 4.3. Localization And Category 111 -- 4.4. Category And The Mislin Genus 114 -- 4.5. Fibrewise Construction 120 -- 4.6. Fibrewise Construction And Category 121 -- 4.7. Examples Of Fibrewise Construction 123 -- Chapter 5. Rational Homotopy And Category 129 -- 5.2. Rational Homotopy Theory 130 -- 5.3. Rational Category And Minimal Models 137 -- 5.4. Rational Category And Fibrations, Including Products 144 -- 5.5. Lower And Upper Bounds In The Rational Context 153 -- 5.6. Geometric Version Of Mcat 158 -- Chapter 6. Hopf Invariants 165 -- 6.2. Hopf Invariants Of Maps S[superscript R] To S[superscript N] 167 -- 6.3. The Berstein-hilton Definition 172 -- 6.4. Hopf Invariants And Ls-category 176 -- 6.5. Crude Hopf Invariants 180 -- 6.7. Hopf-ganea Invariants 188 -- 6.8. Iwase's Counterexamples To The Ganea Conjecture 192 -- 6.9. Fibrewise Construction And Hopf Invariants 195 -- Chapter 7. Category And Critical Points 203 -- 7.2. Relative Category 204 -- 7.3. Local Study Of Isolated Critical Points 208 -- 7.4. Functions With Few Critical Points: The Stable Case 213 -- 7.5. Closed Manifolds 217 -- 7.6. Fusion Of Critical Points And Hopf Invariants 221 -- 7.7. Functions Quadratic At Infinity 225 -- Chapter 8. Category And Symplectic Topology 233 -- 8.2. The Arnold Conjecture 233 -- 8.3. Manifolds With [omega Vertical Bar Superscript Pi 2m] = 0 And Category Weight 240 -- 8.4. The Arnold Conjecture For Symplectically Aspherical Manifolds 244 -- 8.5. Other Symplectic Connections 245 -- Chapter 9. Examples, Computations And Extensions 253 -- 9.2. Category And The Free Loop Space 253 -- 9.3. Sectional Category 259 -- 9.4. Category And The Complexity Of Algorithms 263 -- 9.5. Category And Group Actions 267 -- 9.6. Category Of Lie Groups 273 -- 9.7. Category And 3-manifolds 279 -- 9.8. Other Developments 282 -- Appendix A. Topology And Analysis 287 -- A.1. Types Of Spaces 287 -- A.2. Morse Theory 289 -- Appendix B. Basic Homotopy 293 -- B.1. Whitehead's Theorem 293 -- B.2. Homotopy Pushouts And Pullbacks 293 -- B.3. Cofibrations 295 -- B.4. Fibrations 298 -- B.5. Mixing Cofibrations And Fibrations 301 -- B.6. Properties Of Homotopy Pushouts 301 -- B.7. Properties Of Homotopy Pullbacks 302 -- B.8. Mixing Homotopy Pushouts And Homotopy Pullbacks 303 -- B.9. Homotopy Limits And Colimits 306. Octav Cornea ... [et Al.]. Includes Bibliographical References (p. 311-324) And Index. Cover S Title Photos Lusternik-Schnirelmann Category Copyright © 2003 by the American Mathematical Society ISBN 0-8218-3404-5 QA612.L87 2003 514'.2-dc2l LCCN 2003048136 Dedication Contents Preface Mathematical Surveys and Monographs, Vol. 103 CHAPTER 1 Introduction to LS-Category 1.1. Introduction 1.2. The Definition and Basic Properties 1.3. The Lusternik-Schnirelmann Theorem 1.4. Sums, Homotopy Invariance and Mapping Cones 1.5. Products and Fibrations 1.6. The Whitehead and Ganea Formulations of Category 1.7. Axioms and Category 1.7.1. Abstract Category Axioms. 1.7.2. Abstract Strong Category Axioms. Exercises for Chapter 1 CHAPTER 2 Lower Bounds for LS-Category 2.1. Introduction 2.2. Ganea Fibrations of a Product 2.3. Toomer's Invariant 2.4. Weak Category 2.5. Conilpotency of a Suspension 2.6. Suspension of the Category 2.7. Category Weight 2.8. Comparison Theorem 2.9. Examples Exercises for Chapter 2 CHAPTER 3 Upper Bounds for Category 3.1. Introduction 3.2. First Properties of Upper Bounds 3.3. Geometric Category is not a Homotopy Invariant 3.4. Strong Category and Category Differ by at Most One 3.5. Cone-length 3.6. Stabilization of Ball Category 3.7. Constraints Implying Equality of Category and Upper Bounds Exercises for Chapter 3 CHAPTER 4 Localization and Category 4.1. Introduction 4.2. Localization of Groups and Spaces 4.3. Localization and Category 4.4. Category and the Mislin Genus 4.5. Fibrewise Construction 4.6. Fibrewise Construction and Category 4.7. Examples of Fibrewise Construction Exercises for Chapter CHAPTER 5 Rational Homotopy and Category 5.1. Introduction 5.2. Rational Homotopy Theory 5.2.1. Differential Graded Algebras and PL forms 5.2.2. Minimal Models and Spatial Realization 5.2.3. Model for a Fibration. 5.2.4. Model for a Homotopy Pushout 5.3. Rational Category and Minimal Models 5.4. Rational Category and Fibrations, Including Products 5.5. Lower and Upper Bounds in the Rational Context 5.6. Geometric Version of mcat Exercises for Chapter 5 CHAPTER 6 Hopf Invariants 6.1. Introduction 6.2. Hopf Invariants of Maps S^r ---> S^n 6.3. The Berstein-Hilton Definition 6.4. Hopf Invariants and LS-category 6.5. Crude Hopf Invariants 6.6. Examples 6.7. Hopf-Ganea Invariants 6.8. Iwase's Counterexamples to the Ganea Conjecture 6.9. Fibrewise Construction and Hopf Invariants Exercises for Chapter 6 CHAPTER 7 Category and Critical Points 7.1. Introduction 7.2. Relative Category 7.3. Local Study of Isolated Critical Points 7.4. Functions with Few Critical Points: the Stable Case 7.5. Closed Manifolds 7.6. Fusion of Critical Points and Hopf Invariants 7.7. Functions Quadratic at Infinity Exercises for Chapter 7 CHAPTER 8 Category and Symplectic Topology 8.1. Introduction 8.2. The Arnold Conjecture 8.3. Manifolds with wl,.2n,1 = 0 and Category Weight 8.4. The Arnold Conjecture for Symplectically Aspherical Manifolds 8.5. Other Symplectic Connections 8.5.1. The Arnold Conjecture for Lagrangian Intersections 8.5.2. Symplectic Group Actions Exercises for Chapter 8 CHAPTER 9 Examples, Computations and Extensions 9.1. Introduction 9.2. Category and the Free Loop Space 9.2.1. The Fadell-Husseini Approach. 9.2.2. The Mapping Theorem Approach 9.3. Sectional Category 9.4. Category and the Complexity of Algorithms 9.5. Category and Group Actions 9.6. Category of Lie Groups 9.7. Category and 3-Manifolds 9.8. Other Developments Exercises for Chapter 9 APPENDIX A Topology and Analysis A.1. Types of Spaces APPENDIX B Basic Homotopy B.1. Whitehead's Theorem B.2. Homotopy Pushouts and Pullbacks B.3. Cofibrations B.4. Fibrations B.5. Mixing Cofibrations and Fibrations B.6. Properties of Homotopy Pushouts B.7. Properties of Homotopy Pullbacks B.8. Mixing Homotopy Pushouts and Homotopy Pullbacks B.9. Homotopy Limits and Colimits Bibliography Index Back Cover ``Lusternik-Schnirelmann category is like a Picasso painting. Looking at category from different perspectives produces completely different impressions of category's beauty and applicability.'' --from the Introduction Lusternik-Schnirelmann category is a subject with ties to both algebraic topology and dynamical systems. The authors take LS-category as the central theme, and then develop topics in topology and dynamics around it. Included are exercises and many examples. The book presents the material in a rich, expository style. The book provides a unified approach to LS-category, including foundational material on homotopy theoretic aspects, the Lusternik-Schnirelmann theorem on critical points, and more advanced topics such as Hopf invariants, the construction of functions with few critical points, connections with symplectic geometry, the complexity of algorithms, and category of $3$-manifolds. This is the first book to synthesize these topics. It takes readers from the very basics of the subject to the state of the art. Prerequisites are two semesters of algebraic topology and, perhaps, differential topology. It is suitable for graduate students and researchers interested in algebraic topology and dynamical systems. Lusternik-Schnirelmann(LS) category is a subject with ties to both algebraic topology and dynamical systems. This book takes LS-category as the central theme, and then develops topics in topology and dynamics around it. It is suitable for graduate students and researchers interested in algebraic topology and dynamical systems.