| Preface Notation and Terminology CHAPTER I Two-Dimensional Manifolds 1. Introduction 2. Definition and Examples of n-Manifolds 3. Orientable vs. Nonorientable Manifolds 4. Examples of Compact, Connected 2-Manifolds 5. Statement of the Classification Theorem for Compact Surfaces 6. Triangulations of Compact Surfaces 7. Proof of Theorem 5.1 8. The Euler Characteristic of a Surface References CHAPTER II The Fundamental Group 1. Introduction 2. Basic Notation and Terminology 3. Definition of the Fundamental Group of a Space 4. The Effect of a Continuous Mapping on the Fundamental Group 5. The Fundamental Group of a Circle Is Infinite Cyclic 6. Application: The Brouwer Fixed-Point Theorem in Dimension 2 7. The Fundamental Group of a Product Space 8. Homotopy Type and Homotopy Equivalence of Spaces References CHAPTER III Free Groups and Free Products of Groups 1. Introduction 2. The Weak Product of Abelian Groups 3. Free Abelian Groups 4. Free Products of Groups 5. Free Groups 6. The Presentation of Groups by Generators and Relations 7. Universal Mapping Problems References CHAPTER IV Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces. Applications 1. Introduction 2. Statement and Proof of the Theorem of Seifert and Van Kampen 3. First Application of Theorem 2.1 4. Second Application of Theorem 2.1 5. Structure of the Fundamental Group of a Compact Surface 6. Application to Knot Theory 7. Proof of Lemma 2.4 References CHAPTER V Covering Spaces 1. Introduction 2. Definition and Some Examples of Covering Spaces 3. Lifting of Paths to a Covering Space 4. The Fundamental Group of a Covering Space 5. Lifting of Arbitrary Maps to a Covering Space 6. Homomorphisms and Automorphisms of Covering Space …… CHAPTER VI CHAPTER VII CHAPTER VIII CHAPTER IX CHAPTER X CHAPTER XI CHAPTER XII CHAPTER XIII CHAPTER XIV CHAPTER XV APPENDIX A APPENDIX B Index |
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