"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.
Author(s): Octav Cornea
Series: Mathematical Surveys and Monographs, Vol. 103
Publisher: American Mathematical Society
Year: 2003
Language: English
Pages: C+XVIII+330+B
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
