Algebraic Topology

ISBN-10: 0521795400
ISBN-13: 9780521795401
Edition: 2002
Authors: Allen Hatcher
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Description: 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  More...

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Book details

List price: $49.99
Copyright year: 2002
Publisher: Cambridge University Press
Publication date: 12/3/2001
Binding: Paperback
Pages: 556
Size: 6.97" wide x 9.96" long x 1.26" tall
Weight: 2.420
Language: English

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.

Preface
Standard Notations
Some Underlying Geometric Notions
Homotopy and Homotopy Type
Cell Complexes
Operations on Spaces
Two Criteria for Homotopy Equivalence
The Homotopy Extension Property
The Fundamental Group
Basic Constructions
Paths and Homotopy
The Fundamental Group of the Circle
Induced Homomorphisms
Van Kampen's Theorem
Free Products of Groups
The van Kampen Theorem
Applications to Cell Complexes
Covering Spaces
Lifting Properties
The Classification of Covering Spaces
Deck Transformations and Group Actions
Additional Topics
Graphs and Free Groups
K(G,1) Spaces and Graphs of Groups
Homology
Simplicial and Singular Homology
[Delta]-Complexes
Simplicial Homology
Singular Homology
Homotopy Invariance
Exact Sequences and Excision
The Equivalence of Simplicial and Singular Homology
Computations and Applications
Degree
Cellular Homology
Mayer-Vietoris Sequences
Homology with Coefficients
The Formal Viewpoint
Axioms for Homology
Categories and Functors
Additional Topics
Homology and Fundamental Group
Classical Applications
Simplicial Approximation
Cohomology
Cohomology Groups
The Universal Coefficient Theorem
Cohomology of Spaces
Cup Product
The Cohomology Ring
A Kunneth Formula
Spaces with Polynomial Cohomology
Poincare Duality
Orientations and Homology
The Duality Theorem
Connection with Cup Product
Other Forms of Duality
Additional Topics
Universal Coefficients for Homology
The General Kunneth Formula
H-Spaces and Hopf Algebras
The Cohomology of SO(n)
Bockstein Homomorphisms
Limits and Ext
Transfer Homomorphisms
Local Coefficients
Homotopy Theory
Homotopy Groups
Definitions and Basic Constructions
Whitehead's Theorem
Cellular Approximation
CW Approximation
Elementary Methods of Calculation
Excision for Homotopy Groups
The Hurewicz Theorem
Fiber Bundles
Stable Homotopy Groups
Connections with Cohomology
The Homotopy Construction of Cohomology
Fibrations
Postnikov Towers
Obstruction Theory
Additional Topics
Basepoints and Homotopy
The Hopf Invariant
Minimal Cell Structures
Cohomology of Fiber Bundles
The Brown Representability Theorem
Spectra and Homology Theories
Gluing Constructions
Eckmann-Hilton Duality
Stable Splittings of Spaces
The Loopspace of a Suspension
The Dold-Thom Theorem
Steenrod Squares and Powers
Appendix
Topology of Cell Complexes
The Compact-Open Topology
Bibliography
Index

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