Algebraic Topology

Algebraic Topology - Allen Hatcher Copyright c 2001 by...

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Unformatted text preview: Allen Hatcher Copyright c 2001 by Allen Hatcher Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Standard Notations xii. Chapter 0. Some Underlying Geometric Notions . . . . . 1 Homotopy and Homotopy Type 1. Cell Complexes 5. Operations on Spaces 8. Two Criteria for Homotopy Equivalence 10. The Homotopy Extension Property 14. Chapter 1. The Fundamental Group . . . . . . . . . . . . . 21 1.1. Basic Constructions . . . . . . . . . . . . . . . . . . . . . 25 Paths and Homotopy 25. The Fundamental Group of the Circle 28. Induced Homomorphisms 33. 1.2. Van Kampens Theorem . . . . . . . . . . . . . . . . . . . 38 Free Products of Groups 39. The van Kampen Theorem 41. Applications to Cell Complexes 48. 1.3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 54 Lifting Properties 58. The Classication of Covering Spaces 61. Deck Transformations and Group Actions 68. Additional Topics 1 . A . Graphs and Free Groups 81. 1 . B . K(G,1) Spaces and Graphs of Groups 85. vi Table of Contents Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 95 2.1. Simplicial and Singular Homology . . . . . . . . . . . . . 100 Complexes 100. Simplicial Homology 102. Singular Homology 106. Homotopy Invariance 108. Exact Sequences and Excision 111. The Equivalence of Simplicial and Singular Homology 126. 2.2. Computations and Applications . . . . . . . . . . . . . . 132 Degree 132. Cellular Homology 135. Mayer-Vietoris Sequences 147. Homology with Coefcients 151. 2.3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . 158 Axioms for Homology 158. Categories and Functors 160. Additional Topics 2 . A . Homology and Fundamental Group 164. 2 . B . Classical Applications 167. 2 . C . Simplicial Approximation 175. Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 183 3.1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 188 The Universal Coefcient Theorem 188. Cohomology of Spaces 195. 3.2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . 204 The Cohomology Ring 209. A Kunneth Formula 216. Spaces with Polynomial Cohomology 222. 3.3. Poincar e Duality . . . . . . . . . . . . . . . . . . . . . . . . 228 Orientations and Homology 231. The Duality Theorem 237. Connection with Cup Product 247. Other Forms of Duality 250. Additional Topics 3 . A . Universal Coefcients for Homology 259. 3 . B . The General Kunneth Formula 266. 3 . C . HSpaces and Hopf Algebras 279. 3 . D . The Cohomology of SO(n) 290. 3 . E . Bockstein Homomorphisms 301....
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This note was uploaded on 09/23/2010 for the course MATH 1121 taught by Professor Dr.mcgrawhill during the Spring '10 term at SUNY Buffalo.

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Algebraic Topology - Allen Hatcher Copyright c 2001 by...

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