In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.
Author(s): Allen Hatcher
Edition: 1
Publisher: Cambridge University Press
Year: 2001
Language: English
Commentary: Free Web version (15/10/2017), bookmarked
Pages: 556
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 29.
Induced Homomorphisms 34.
1.2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . 40
Free Products of Groups 41. The van Kampen Theorem 43.
Applications to Cell Complexes 49.
1.3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . 56
Lifting Properties 60. The Classification of Covering Spaces 63.
Deck Transformations and Group Actions 70.
Additional Topics
1.A. Graphs and Free Groups 83.
1.B. K(G,1) Spaces and Graphs of Groups 87.
Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 97
2.1. Simplicial and Singular Homology . . . . . . . . . . . . . 102
Δ Complexes 102. Simplicial Homology 104. Singular Homology 108.
Homotopy Invariance 110. Exact Sequences and Excision 113.
The Equivalence of Simplicial and Singular Homology 128.
2.2. Computations and Applications . . . . . . . . . . . . . . 134
Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149.
Homology with Coefficients 153.
2.3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . 160
Axioms for Homology 160. Categories and Functors 162.
Additional Topics
2.A. Homology and Fundamental Group 166.
2.B. Classical Applications 169.
2.C. Simplicial Approximation 177.
Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 185
3.1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 190
The Universal Coefficient Theorem 190. Cohomology of Spaces 197.
3.2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . 206
The Cohomology Ring 212. A K¨unneth Formula 214.
Spaces with Polynomial Cohomology 220.
3.3. Poincar´e Duality . . . . . . . . . . . . . . . . . . . . . . . . 230
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¨unneth Formula 268.
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 . . . . . . . . . . . . . . . . . 337
4.1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . 339
Definitions and Basic Constructions 340. Whitehead’s Theorem 346.
Cellular Approximation 348. CW Approximation 352.
4.2. Elementary Methods of Calculation . . . . . . . . . . . . 360
Excision for Homotopy Groups 360. The Hurewicz Theorem 366.
Fiber Bundles 375. Stable Homotopy Groups 384.
4.3. Connections with Cohomology . . . . . . . . . . . . . . 393
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. The Dold-Thom Theorem 475.
4.L. Steenrod Squares and Powers 487.
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Topology of Cell Complexes 519. The Compact-Open Topology 529.
The Homotopy Extension Property 532. Simplicial CW Structures 533.
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
