Pub. Date:
Cambridge University Press
Algebraic Topology

Algebraic Topology

by Allen Hatcher
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Product Details

ISBN-13: 9780521541862
Publisher: Cambridge University Press
Publication date: 09/01/2005
Product dimensions: 6.00(w) x 1.25(h) x 9.00(d)

Table of Contents

Standard Notationsxii
Chapter 0.Some Underlying Geometric Notions1
Homotopy and Homotopy Type1
Cell Complexes5
Operations on Spaces8
Two Criteria for Homotopy Equivalence10
The Homotopy Extension Property14
Chapter 1.The Fundamental Group21
1.1.Basic Constructions25
Paths and Homotopy25
The Fundamental Group of the Circle29
Induced Homomorphisms34
1.2.Van Kampen's Theorem40
Free Products of Groups41
The van Kampen Theorem43
Applications to Cell Complexes50
1.3.Covering Spaces56
Lifting Properties60
The Classification of Covering Spaces63
Deck Transformations and Group Actions70
Additional Topics
1.A.Graphs and Free Groups83
1.B.K(G,1) Spaces and Graphs of Groups87
Chapter 2.Homology97
2.1.Simplicial and Singular Homology102
Simplicial Homology104
Singular Homology108
Homotopy Invariance110
Exact Sequences and Excision113
The Equivalence of Simplicial and Singular Homology128
2.2.Computations and Applications134
Cellular Homology137
Mayer-Vietoris Sequences149
Homology with Coefficients153
2.3.The Formal Viewpoint160
Axioms for Homology160
Categories and Functors162
Additional Topics
2.A.Homology and Fundamental Group166
2.B.Classical Applications169
2.C.Simplicial Approximation177
Chapter 3.Cohomology185
3.1.Cohomology Groups190
The Universal Coefficient Theorem190
Cohomology of Spaces197
3.2.Cup Product206
The Cohomology Ring211
A Kunneth Formula218
Spaces with Polynomial Cohomology224
3.3.Poincare Duality230
Orientations and Homology233
The Duality Theorem239
Connection with Cup Product249
Other Forms of Duality252
Additional Topics
3.A.Universal Coefficients for Homology261
3.B.The General Kunneth Formula268
3.C.H-Spaces and Hopf Algebras281
3.D.The Cohomology of SO(n)292
3.E.Bockstein Homomorphisms303
3.F.Limits and Ext311
3.G.Transfer Homomorphisms321
3.H.Local Coefficients327
Chapter 4.Homotopy Theory337
4.1.Homotopy Groups339
Definitions and Basic Constructions340
Whitehead's Theorem346
Cellular Approximation348
CW Approximation352
4.2.Elementary Methods of Calculation360
Excision for Homotopy Groups360
The Hurewicz Theorem366
Fiber Bundles375
Stable Homotopy Groups384
4.3.Connections with Cohomology393
The Homotopy Construction of Cohomology393
Postnikov Towers410
Obstruction Theory415
Additional Topics
4.A.Basepoints and Homotopy421
4.B.The Hopf Invariant427
4.C.Minimal Cell Structures429
4.D.Cohomology of Fiber Bundles431
4.E.The Brown Representability Theorem448
4.F.Spectra and Homology Theories452
4.G.Gluing Constructions456
4.H.Eckmann-Hilton Duality460
4.I.Stable Splittings of Spaces466
4.J.The Loopspace of a Suspension470
4.K.The Dold-Thom Theorem475
4.L.Steenrod Squares and Powers487
Topology of Cell Complexes519
The Compact-Open Topology529

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From the Publisher

"Algebraic topoligy books that emphasize geometrical intuition usually have only a modest technical reach. Remarkably, Hatcher (Cornell Univ.) offers a highly geometrical treatment that neverheless matches the coverage of, e.g., Edwin Henry Spanier's very formidable and identically titled classic work... He promises two advanced companion volumes, one on spectral sequences, one on vector bundles. One anticipates the combined treatise doing for algebraic topology what Michael Spivak's magisterial five-volume set did for differential geometry." Choice

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