Algebraic Topology

Name: Algebraic Topology
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ISBN: 9780521795401

By: Allen Hatcher

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Paperback Published: 8th April 2002
ISBN: 9780521795401
Number Of Pages: 544

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In most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology. This introductory textbook in algebraic topology is 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. The four main chapters present the basic material of the subject: 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 of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find. Among these are: 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 a full exposition of Steenrod squares and powers. Researchers will also welcome this aspect of the book.

Industry Reviews

' ... this is a marvellous tome, which is indeed a delight to read. This book is destined to become very popular amongst students and teachers alike.' Bulletin of the Belgian Mathematical Society '... clear and concise ... makes the book useful both as a basis for courses and as a reference work.' Monatshefte fur Mathematik '... the truly unusual abundance of instructive examples and complementing exercises is absolutely unique of such a kind ... the distinctly circumspect, methodologically inductive, intuitive, descriptively elucidating and very detailed style of writing give evidence to the fact that the author's first priorities are exactly what students need when working with such a textbook, namely clarity, readability, steady motivation, guided inspiration, increasing demand, and as much self-containedness of the exposition as possible. No doubt, a very devoted and experienced teacher has been at work here, very much so to the benefit of beginners in the field of algebraic topology, instructors, and interested readers in general.' Zentralblatt MATH

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
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521795401
ISBN-10: 0521795400
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 544
Published: 8th April 2002
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 25.4 x 17.78 x 3.81
Weight (kg): 0.95