代数拓扑学1.pdf

Allen Hatcher
Copyright c 2002 by Cambridge University Press
Single paper or electronic copies for mercial personal use may be made without explicit permission from the
author or publisher. All other rights reserved.
Preface ..................................ix
Standard Notations xii.
Chapter 0. Some Underlying Geometric Notions ..... 1
Homotopy and Homotopy Type 1. plexes 5.
Operations on Spaces 8. Two Criteria for Homotopy Equivalence 10.
The Homotopy Extension Property 14.
Chapter 1. The Fundamental Group ............. 21
. Basic Constructions ..................... 25
Paths and Homotopy 25. The Fundamental Group of the Circle 29.
Induced Homomorphisms 34.
. Van Kampen’s Theorem ................... 40
Free Products of Groups 41. The van Kampen Theorem 43.
Applications to plexes 50.
. Covering Spaces ........................ 56
Lifting Properties 60. The Classification of Covering Spaces 63.
Deck Transformations and Group Actions 70.
Additional Topics
. Graphs and Free Groups 83.
. K(G,1) Spaces and Graphs of Groups 87.
Chapter 2. Homology ....................... 97
. Simplicial and Singular Homology .............102
∆Complexes 102. Simplicial Homology 104. Singular Homology 108.
Homotopy Invariance 110. Exact Sequences and Excision 113.
The Equivalence of Simplicial and Singular Homology 128.
. Computations and Applications ..............134
Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149.
Homology with Coefficients 153.
. The Formal Viewpoint ....................160
Axioms for Homology 160. Categories and Functors 162.
Additional Topics
. Homology and Fundamental Group 166.
. Classical Applications 169.
. Simplicial Approximation 177.
Chapter 3. Cohomology .....................185
. Cohomology Groups .....................190
The Universal Coefficient Theorem 190. Cohomology of Spaces 197.
. Cup Product ..........................206
The Co