Lusternik-Schnirelmann category

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Lusternik-Schnirelmann Category

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     © 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

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