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Algebraic Topology
Contents:
Preface
Standard Notations
Chapter 0. Some Underlying Geometric Notations
Homotopy and Homotopy Type
Cell complexes
Operations on Spaces
Two Criteria for Homotopy Equivalence
The Homotopy Extension Property
Exercises
Chapter 1. The Fundamental Group
1.1. Basic Constructions
Paths and Homotopy
The Fundamental Group of the Circle
Induced Homomorphisms
Exercises
1.2. Van Kampen’s Theorem
Free Products of Groups
The van Kampen theorem
Applications to Cell Complexes
Exercises
1.3. Covering Spaces
Lifting Properties
The Classification of Covering Spaces
Deck Transformations and Group Actions
Exercises
Additional Topics
1.A. Graphs and Free Groups
1.B. K(G,1) Spaces and Graphs of Groups
Chapter 2. Homology
2.1. Simplicial and Singular Homology
\(\Delta\)
-Complexes
Simplicial Homology
Singular Homology
Homotopy Invariance
Exact Sequences and Excision
The Equivalence of Simplicial and Singular Homology
2.2. Computations and Applications
Degree
Cellular Homology
Mayer Vietoris Sequences
Homology with Coefficients
2.3. The Formal Viewpoint
Axioms for Homology
Categories and Functors
Additional Topics
2.A. Homology and Fundamental Group
2.B. Classical Applications
2.C. Simplicial Approximation
Chapter 3. Cohomology
3.1. Cohomology Groups
The Universal Coefficient Theorem
Cohomology of Spaces
3.2. Cup Product
The Cohomology Ring
A Künneth Formula
Spaces with Polynomial Cohomology
3.3. Poincaré Duality
Orientations and Homology
The Duality Theorem
Connection with Cup Product
Other Forms of Duality
Additional Topics
3.A. Universal Coefficients for Homology
3.B. The General Künneth Formula
3.C. H-Spaces and Hopf Algebras
3.D. The Cohomology of SO(n)
3.E. Bockstein Homomorphisms
3.F. Limits and Ext
3.G. Transfer Homomorphisms
3.H. Local Coefficients
Chapter 4. Homotopy Theory
4.1. Homotopy Groups
Definitions and Basic Constructions
Whitehead’s Theorem
Cellular Approximation
CW Approximation
4.2. Elementary Methods of Calculation
Excision for Homotopy Groups
The Hurewicz Theorem
Fiber Bundles
Stable Homotopy Groups
4.3. Connections with Cohomology
The Homotopy Construction of Cohomology
Fibrations
Postnikov Towers
Obstruction Theory
Additional Topics
4.A. Basepoints and Homotopy
4.B. The Hopf Invariant
4.C. Minimal Cell Structures
4.D. Cohomology of Fiber Bundles
4.E. The Brown Representability Theorem
4.F. Spectra and Homology Theories
4.G. Gluing Constructions
4.H. Eckmann-Hilton Duality
4.I. Stable Splittings of Spaces
4.J. The Loopspace of a Suspension
4.K. The Dold-Thom Theorem
4.L. Steenrod Squares and Powers
Appendix
Topology of Cell Complexes
The Compact-Open Topology
The Homotopy Extension Property
Simplicial CW Structures
Bibliography
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3.D. The Cohomology of SO(n)
3.D. The Cohomology of SO(n)
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