简介
Summary:
Publisher Summary 1
Hatcher (mathematics, Cornell U.) presents an introduction to algebraic topology that retains the classical feel of the field from three or four decades ago, but clarifies some of the most important results and techniques that have arisen in the intervening years. The emphasis leans towards the geometric, rather than algebraic, aspects of the subject. After presenting some of the basic geometric concepts and constructions of the subject, Hatcher separates his treatment into the two broad topics of homology and homotopy. He assumes the reader has familiarity with the content of standard courses in algebra and point-set topology. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Publisher Summary 2
An introductory textbook suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.
Publisher Summary 3
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.
目录
Preface p. ix
Standard Notations p. xii
Some Underlying Geometric Notions p. 1
Homotopy and Homotopy Type p. 1
Cell Complexes p. 5
Operations on Spaces p. 8
Two Criteria for Homotopy Equivalence p. 10
The Homotopy Extension Property p. 14
The Fundamental Group p. 21
Basic Constructions p. 25
Paths and Homotopy p. 25
The Fundamental Group of the Circle p. 29
Induced Homomorphisms p. 34
Van Kampen's Theorem p. 40
Free Products of Groups p. 41
The van Kampen Theorem p. 43
Applications to Cell Complexes p. 50
Covering Spaces p. 56
Lifting Properties p. 60
The Classification of Covering Spaces p. 63
Deck Transformations and Group Actions p. 70
Additional Topics
Graphs and Free Groups p. 83
K(G,1) Spaces and Graphs of Groups p. 87
Homology p. 97
Simplicial and Singular Homology p. 102
[Delta]-Complexes p. 102
Simplicial Homology p. 104
Singular Homology p. 108
Homotopy Invariance p. 110
Exact Sequences and Excision p. 113
The Equivalence of Simplicial and Singular Homology p. 128
Computations and Applications p. 134
Degree p. 134
Cellular Homology p. 137
Mayer-Vietoris Sequences p. 149
Homology with Coefficients p. 153
The Formal Viewpoint p. 160
Axioms for Homology p. 160
Categories and Functors p. 162
Additional Topics
Homology and Fundamental Group p. 166
Classical Applications p. 169
Simplicial Approximation p. 177
Cohomology p. 185
Cohomology Groups p. 190
The Universal Coefficient Theorem p. 190
Cohomology of Spaces p. 197
Cup Product p. 206
The Cohomology Ring p. 211
A Kunneth Formula p. 218
Spaces with Polynomial Cohomology p. 224
Poincare Duality p. 230
Orientations and Homology p. 233
The Duality Theorem p. 239
Connection with Cup Product p. 249
Other Forms of Duality p. 252
Additional Topics
Universal Coefficients for Homology p. 261
The General Kunneth Formula p. 268
H-Spaces and Hopf Algebras p. 281
The Cohomology of SO(n) p. 292
Bockstein Homomorphisms p. 303
Limits and Ext p. 311
Transfer Homomorphisms p. 321
Local Coefficients p. 327
Homotopy Theory p. 337
Homotopy Groups p. 339
Definitions and Basic Constructions p. 340
Whitehead's Theorem p. 346
Cellular Approximation p. 348
CW Approximation p. 352
Elementary Methods of Calculation p. 360
Excision for Homotopy Groups p. 360
The Hurewicz Theorem p. 366
Fiber Bundles p. 375
Stable Homotopy Groups p. 384
Connections with Cohomology p. 393
The Homotopy Construction of Cohomology p. 393
Fibrations p. 405
Postnikov Towers p. 410
Obstruction Theory p. 415
Additional Topics
Basepoints and Homotopy p. 421
The Hopf Invariant p. 427
Minimal Cell Structures p. 429
Cohomology of Fiber Bundles p. 431
The Brown Representability Theorem p. 448
Spectra and Homology Theories p. 452
Gluing Constructions p. 456
Eckmann-Hilton Duality p. 460
Stable Splittings of Spaces p. 466
The Loopspace of a Suspension p. 470
The Dold-Thom Theorem p. 475
Steenrod Squares and Powers p. 487
Appendix p. 519
Topology of Cell Complexes p. 519
The Compact-Open Topology p. 529
Bibliography p. 533
Index p. 539
Standard Notations p. xii
Some Underlying Geometric Notions p. 1
Homotopy and Homotopy Type p. 1
Cell Complexes p. 5
Operations on Spaces p. 8
Two Criteria for Homotopy Equivalence p. 10
The Homotopy Extension Property p. 14
The Fundamental Group p. 21
Basic Constructions p. 25
Paths and Homotopy p. 25
The Fundamental Group of the Circle p. 29
Induced Homomorphisms p. 34
Van Kampen's Theorem p. 40
Free Products of Groups p. 41
The van Kampen Theorem p. 43
Applications to Cell Complexes p. 50
Covering Spaces p. 56
Lifting Properties p. 60
The Classification of Covering Spaces p. 63
Deck Transformations and Group Actions p. 70
Additional Topics
Graphs and Free Groups p. 83
K(G,1) Spaces and Graphs of Groups p. 87
Homology p. 97
Simplicial and Singular Homology p. 102
[Delta]-Complexes p. 102
Simplicial Homology p. 104
Singular Homology p. 108
Homotopy Invariance p. 110
Exact Sequences and Excision p. 113
The Equivalence of Simplicial and Singular Homology p. 128
Computations and Applications p. 134
Degree p. 134
Cellular Homology p. 137
Mayer-Vietoris Sequences p. 149
Homology with Coefficients p. 153
The Formal Viewpoint p. 160
Axioms for Homology p. 160
Categories and Functors p. 162
Additional Topics
Homology and Fundamental Group p. 166
Classical Applications p. 169
Simplicial Approximation p. 177
Cohomology p. 185
Cohomology Groups p. 190
The Universal Coefficient Theorem p. 190
Cohomology of Spaces p. 197
Cup Product p. 206
The Cohomology Ring p. 211
A Kunneth Formula p. 218
Spaces with Polynomial Cohomology p. 224
Poincare Duality p. 230
Orientations and Homology p. 233
The Duality Theorem p. 239
Connection with Cup Product p. 249
Other Forms of Duality p. 252
Additional Topics
Universal Coefficients for Homology p. 261
The General Kunneth Formula p. 268
H-Spaces and Hopf Algebras p. 281
The Cohomology of SO(n) p. 292
Bockstein Homomorphisms p. 303
Limits and Ext p. 311
Transfer Homomorphisms p. 321
Local Coefficients p. 327
Homotopy Theory p. 337
Homotopy Groups p. 339
Definitions and Basic Constructions p. 340
Whitehead's Theorem p. 346
Cellular Approximation p. 348
CW Approximation p. 352
Elementary Methods of Calculation p. 360
Excision for Homotopy Groups p. 360
The Hurewicz Theorem p. 366
Fiber Bundles p. 375
Stable Homotopy Groups p. 384
Connections with Cohomology p. 393
The Homotopy Construction of Cohomology p. 393
Fibrations p. 405
Postnikov Towers p. 410
Obstruction Theory p. 415
Additional Topics
Basepoints and Homotopy p. 421
The Hopf Invariant p. 427
Minimal Cell Structures p. 429
Cohomology of Fiber Bundles p. 431
The Brown Representability Theorem p. 448
Spectra and Homology Theories p. 452
Gluing Constructions p. 456
Eckmann-Hilton Duality p. 460
Stable Splittings of Spaces p. 466
The Loopspace of a Suspension p. 470
The Dold-Thom Theorem p. 475
Steenrod Squares and Powers p. 487
Appendix p. 519
Topology of Cell Complexes p. 519
The Compact-Open Topology p. 529
Bibliography p. 533
Index p. 539
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