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Unformatted text preview: Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S 1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert–Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert–Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classiﬁcation Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence– the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π 1 and H 1 126 8. The MayerVietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. The Lefschetz Fixed Point Theorem 158 Chapter 8. Cell Complexes 165 1. Introduction 165 2. Adjunction Spaces 169 3. CW Complexes 176 4. The Homology of CW complexes 177 Chapter 9. Products and the K¨unneth Theorem 185 1. Introduction to the K¨unneth Theorem 185 2. Tensor Products of Chain Complexes 187 3. Tor and the K¨unneth Theorem for Chain Complexes 191 4. Tensor and Tor for Other Rings 198 5. Homology with Coeﬃcients 199 6. The EilenbergZilber Theorem 201 Chapter 10. Cohomology 209 1. Cohomology 209 2. The Universal Coeﬃcient Theorem 214 3. Cup Products 218 4. Calculation of Cup Products 222 Chapter 11. Manifolds and Poincar´ e duality 227 1. Manifolds 227 2. Poincar´ e Duality 232 3. Applications of Poincar´ e Duality 235 4. Cohomology with Compact Supports 239 5. Proof of Poincar´ e Duality 242 CHAPTER 1 Introduction 1. Introduction Topology is the study of properties of topological spaces invariant under homeomorphisms. See Section 2 for a precise deﬁnition of topological space. In algebraic topology, one tries to attach algebraic invariants to spaces and to maps of spaces which allow us to use algebra, which is usually simpler, rather than geometry. (But, the underlying motivation is to solve geometric problems.) A simple example is the use of the Euler characteristic to distinguish closed surfaces. The Euler characteristic is deﬁned as follows. Imagine the surface (say a sphere in...
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This note was uploaded on 10/28/2011 for the course MATH 795 taught by Professor Thompson during the Spring '11 term at CUNY Hunter.
 Spring '11
 Thompson
 Algebra, Algebraic Topology

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