algebraic topology

Allen Hatcher

Copyright c 2002 by Cambridge University Press
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Preface

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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
1.1. Basic Constructions

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21

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25

Paths and Homotopy 25. The Fundamental Group of the Circle 29.
Induced Homomorphisms34.

1.2. Van Kampen’s Theorem

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40

Free Products of Groups 41. The van Kampen Theorem 43.
Applications to Cell Complexes 50.

1.3. Covering Spaces

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Lifting Properties 60. The Classification of Covering Spaces 63.
Deck Transformations and Group Actions 70.

Additional Topics
1.A. Graphs andFree Groups 83.
1.B. K(G,1) Spaces and Graphs of Groups 87.

56

Chapter 2. Homology

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2.1. Simplicial and Singular Homology

97

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∆ Complexes 102. Simplicial Homology 104. Singular Homology 108.
Homotopy Invariance 110. Exact Sequences and Excision 113.
The Equivalence of Simplicial and SingularHomology 128.

2.2. Computations and Applications

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Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149.
Homology with Coefficients 153.

2.3. The Formal Viewpoint

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Axioms for Homology 160. Categories and Functors 162.

Additional Topics
2.A. Homology and Fundamental Group 166.
2.B. ClassicalApplications 169.
2.C. Simplicial Approximation 177.

Chapter 3. Cohomology

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3.1. Cohomology Groups

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The Universal Coefficient Theorem 190. Cohomology of Spaces 197.

3.2. Cup Product

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The Cohomology Ring 211. A K¨nneth Formula 218.
uSpaces with Polynomial Cohomology 224.

3.3. Poincar´ Duality
e

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Orientations and Homology 233. The Duality Theorem 239.
Connection with Cup Product 249. Other Forms of Duality 252.

Additional Topics
3.A. Universal Coefficients for Homology 261.
3.B. The General K¨nneth Formula 268.
u
3.C. H–Spaces and Hopf Algebras 281.3.D. The Cohomology of SO(n) 292.
3.E. Bockstein Homomorphisms 303.
3.F. Limits and Ext 311.
3.G. Transfer Homomorphisms 321.
3.H. Local Coefficients 327.

Chapter 4. Homotopy Theory
4.1. Homotopy Groups

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Definitions and Basic Constructions 340. Whitehead’s Theorem 346.
Cellular Approximation348. CW Approximation 352.

4.2. Elementary Methods of Calculation

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Excision for Homotopy Groups 360. The Hurewicz Theorem 366.
Fiber Bundles 375. Stable Homotopy Groups 384.

4.3. Connections with Cohomology

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The Homotopy Construction of Cohomology 393. Fibrations 405.
Postnikov Towers 410. Obstruction Theory 415.Additional Topics
4.A. Basepoints and Homotopy 421.
4.B. The Hopf Invariant 427.
4.C. Minimal Cell Structures 429.
4.D. Cohomology of Fiber Bundles 431.
4.E. The Brown Representability Theorem 448.
4.F. Spectra and Homology Theories 452.
4.G. Gluing Constructions 456.
4.H. Eckmann-Hilton Duality 460.
4.I.

Stable Splittings of Spaces 466.

4.J. The Loopspace of a Suspension 470.
4.K....