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Algebraic Topology - Allen Hatcher

Algebraic Topology

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

Prefacep. ix
Standard Notationsp. xii
Some Underlying Geometric Notionsp. 1
Homotopy and Homotopy Typep. 1
Cell Complexesp. 5
Operations on Spacesp. 8
Two Criteria for Homotopy Equivalencep. 10
The Homotopy Extension Propertyp. 14
The Fundamental Groupp. 21
Basic Constructionsp. 25
Paths and Homotopyp. 25
The Fundamental Group of the Circlep. 29
Induced Homomorphismsp. 34
Van Kampen's Theoremp. 40
Free Products of Groupsp. 41
The van Kampen Theoremp. 43
Applications to Cell Complexesp. 50
Covering Spacesp. 56
Lifting Propertiesp. 60
The Classification of Covering Spacesp. 63
Deck Transformations and Group Actionsp. 70
Additional Topics
Graphs and Free Groupsp. 83
K(G,1) Spaces and Graphs of Groupsp. 87
Homologyp. 97
Simplicial and Singular Homologyp. 102
[Delta]-Complexesp. 102
Simplicial Homologyp. 104
Singular Homologyp. 108
Homotopy Invariancep. 110
Exact Sequences and Excisionp. 113
The Equivalence of Simplicial and Singular Homologyp. 128
Computations and Applicationsp. 134
Degreep. 134
Cellular Homologyp. 137
Mayer-Vietoris Sequencesp. 149
Homology with Coefficientsp. 153
The Formal Viewpointp. 160
Axioms for Homologyp. 160
Categories and Functorsp. 162
Additional Topics
Homology and Fundamental Groupp. 166
Classical Applicationsp. 169
Simplicial Approximationp. 177
Cohomologyp. 185
Cohomology Groupsp. 190
The Universal Coefficient Theoremp. 190
Cohomology of Spacesp. 197
Cup Productp. 206
The Cohomology Ringp. 211
A Kunneth Formulap. 218
Spaces with Polynomial Cohomologyp. 224
Poincare Dualityp. 230
Orientations and Homologyp. 233
The Duality Theoremp. 239
Connection with Cup Productp. 249
Other Forms of Dualityp. 252
Additional Topics
Universal Coefficients for Homologyp. 261
The General Kunneth Formulap. 268
H-Spaces and Hopf Algebrasp. 281
The Cohomology of SO(n)p. 292
Bockstein Homomorphismsp. 303
Limits and Extp. 311
Transfer Homomorphismsp. 321
Local Coefficientsp. 327
Homotopy Theoryp. 337
Homotopy Groupsp. 339
Definitions and Basic Constructionsp. 340
Whitehead's Theoremp. 346
Cellular Approximationp. 348
CW Approximationp. 352
Elementary Methods of Calculationp. 360
Excision for Homotopy Groupsp. 360
The Hurewicz Theoremp. 366
Fiber Bundlesp. 375
Stable Homotopy Groupsp. 384
Connections with Cohomologyp. 393
The Homotopy Construction of Cohomologyp. 393
Fibrationsp. 405
Postnikov Towersp. 410
Obstruction Theoryp. 415
Additional Topics
Basepoints and Homotopyp. 421
The Hopf Invariantp. 427
Minimal Cell Structuresp. 429
Cohomology of Fiber Bundlesp. 431
The Brown Representability Theoremp. 448
Spectra and Homology Theoriesp. 452
Gluing Constructionsp. 456
Eckmann-Hilton Dualityp. 460
Stable Splittings of Spacesp. 466
The Loopspace of a Suspensionp. 470
The Dold-Thom Theoremp. 475
Steenrod Squares and Powersp. 487
Appendixp. 519
Topology of Cell Complexesp. 519
The Compact-Open Topologyp. 529
Bibliographyp. 533
Indexp. 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

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