Contents Preface xi Chapter 1. Introduction to LS-Category 1 1.1. Introduction 1 1.2. The Definition and Basic Properties 1 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 Exercises for Chapter 1 40 Chapter 2. Lower Bounds for LS-Category 47 2.1. Introduction 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 2.9. Examples 70 Exercises for Chapter 2 71 Chapter 3. Upper Bounds for Category 75 3.1. Introduction 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 Exercises for Chapter 3 101 Chapter 4. Localization and Category 105 4.1. Introduction 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
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