**Mathematical Surveys and Monographs**

Volume: 103;
2003;
330 pp;
Hardcover

MSC: Primary 55; 57; 58; 37; 53;

Print ISBN: 978-0-8218-3404-6

Product Code: SURV/103

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1330-9
Product Code: SURV/103.E**

List Price: $97.00

Individual Member Price: $77.60

#### Supplemental Materials

# Lusternik-Schnirelmann Category

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*Octav Cornea; Gregory Lupton; John Oprea; Daniel Tanré*

“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 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.

#### Reviews & Endorsements

Finally! The book that sums up the explosive development of the Ljusternik-Schnirelman theory in the past decade has now appeared.

-- Zentralblatt MATH

A carefully written, well-conceived and timely addition to the literature on category … copious references, many interesting exercises and two helpful appendices … should prove invaluable both as a reference for experts and as a text for a graduate seminar.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lusternik-Schnirelmann Category

- Contents vii8 free
- Preface xi12 free
- Chapter 1. Introduction to LS-Category 120 free
- 1.1. Introduction 120
- 1.2. The Definition and Basic Properties 120
- 1.3. The Lusternik-Schnirelmann Theorem 726
- 1.4. Sums, Homotopy Invariance and Mapping Cones 1332
- 1.5. Products and Fibrations 1736
- 1.6. The Whitehead and Ganea Formulations of Category 2241
- 1.7. Axioms and Category 3352
- Exercises for Chapter 1 4059

- Chapter 2. Lower Bounds for LS-Category 4766
- Chapter 3. Upper Bounds for Category 7594
- 3.1. Introduction 7594
- 3.2. First Properties of Upper Bounds 7695
- 3.3. Geometric Category is not a Homotopy Invariant 7998
- 3.4. Strong Category and Category Differ by at Most One 82101
- 3.5. Cone-length 83102
- 3.6. Stabilization of Ball Category 92111
- 3.7. Constraints Implying Equality of Category and Upper Bounds 98117
- Exercises for Chapter 3 101120

- Chapter 4. Localization and Category 105124
- 4.1. Introduction 105124
- 4.2. Localization of Groups and Spaces 106125
- 4.3. Localization and Category 111130
- 4.4. Category and the Mislin Genus 114133
- 4.5. Fibrewise Construction 120139
- 4.6. Fibrewise Construction and Category 121140
- 4.7. Examples of Fibrewise Construction 123142
- Exercises for Chapter 4 125144

- Chapter 5. Rational Homotopy and Category 129148
- 5.1. Introduction 129148
- 5.2. Rational Homotopy Theory 130149
- 5.3. Rational Category and Minimal Models 137156
- 5.4. Rational Category and Fibrations, Including Products 144163
- 5.5. Lower and Upper Bounds in the Rational Context 153172
- 5.6. Geometric Version of meat 158177
- Exercises for Chapter 5 161180

- Chapter 6. Hopf Invariants 165184
- 6.1. Introduction 165184
- 6.2. Hopf Invariants of Maps S[sup(r)] → S[sup(n)] 167186
- 6.3. The Berstein-Hilton Definition 172191
- 6.4. Hopf Invariants and LS-category 176195
- 6.5. Crude Hopf Invariants 180199
- 6.6. Examples 184203
- 6.7. Hopf-Ganea Invariants 188207
- 6.8. Iwase's Counterexamples to the Ganea Conjecture 192211
- 6.9. Fibrewise Construction and Hopf Invariants 195214
- Exercises for Chapter 6 199218

- Chapter 7. Category and Critical Points 203222
- 7.1. Introduction 203222
- 7.2. Relative Category 204223
- 7.3. Local Study of Isolated Critical Points 208227
- 7.4. Functions with Few Critical Points: the Stable Case 213232
- 7.5. Closed Manifolds 217236
- 7.6. Fusion of Critical Points and Hopf Invariants 221240
- 7.7. Functions Quadratic at Infinity 225244
- Exercises for Chapter 7 231250

- Chapter 8. Category and Symplectic Topology 233252
- Chapter 9. Examples, Computations and Extensions 253272
- 9.1. Introduction 253272
- 9.2. Category and the Free Loop Space 253272
- 9.3. Sectional Category 259278
- 9.4. Category and the Complexity of Algorithms 263282
- 9.5. Category and Group Actions 267286
- 9.6. Category of Lie Groups 273292
- 9.7. Category and 3-Manifolds 279298
- 9.8. Other Developments 282301
- Exercises for Chapter 9 283302

- Appendix A. Topology and Analysis 287306
- Appendix B. Basic Homotopy 293312
- B.1. Whitehead's Theorem 293312
- B.2. Homotopy Pushouts and Pullbacks 293312
- B.3. Cofibrations 295314
- B.4. Fibrations 298317
- B.5. Mixing Cofibrations and Fibrations 301320
- B.6. Properties of Homotopy Pushouts 301320
- B.7. Properties of Homotopy Pullbacks 302321
- B.8. Mixing Homotopy Pushouts and Homotopy Pullbacks 303322
- B.9. Homotopy Limits and Colimits 306325

- Bibliography 311330
- Index 325344