Algebraic Topology

Table of Contents
Preface
	Standard Notations
Chapter 0. Some Underlying Geometric Notions
	Homotopy and Homotopy Type
	Cell Complexes
	Operations on Spaces
	Two Criteria for Homotopy Equivalence
	The Homotopy Extension Property
Chapter 1. The Fundamental Group
	1.1. Basic Constructions
		Paths and Homotopy
		The Fundamental Group of the Circle
		Induced Homomorphisms
	1.2. Van Kampen’s Theorem
		Free Products of Groups
		The van Kampen Theorem
		Applications to Cell Complexes
	1.3. Covering Spaces
		Lifting Properties
		The Classification of Covering Spaces
		Deck Transformations and Group Actions
	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
		∆ 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. Poincare 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
Index