Lectures on Algebraic Topology

Cover
Title page
Foreword
Chapter I Preliminaries on Categories, Abelian Groups, and Homotopy
	1 Categories and Functors
	2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups)
	3 Homotopy
Chapter II Homology of Complexes
	1 Complexes
	2 Connecting Homomorphism, Exact Homology Sequence
	3 Chain-Homotopy
	4 Free Complexes
Chapter III Singular Homology
	1 Standard Simplices and Their Linear Maps
	2 The Singular Complex
	3 Singular Homology
	4 Special Cases
	5 Invariance under Homotopy
	6 Barycentric Subdivision
	7 Small Simplices. Excision
	8 Mayer-Vietoris Sequences
Chapter IV Applications to Euclidean Space
	1 Standard Maps between Cells and Spheres
	2 Homology of Cells and Spheres
	3 Local Homology
	4 The Degree of a Map
	5 Local Degrees
	6 Homology Properties of Neighborhood Retracts in R^n
	7 Jordan Theorem, Invariance of Domain
	8 Euclidean Neighborhood Retracts (ENRs)
Chapter V Cellular Decomposition and Cellular Homology
	1 Cellular Spaces
	2 CW-Spaces
	3 Examples
	4 Homology Properties of CW-Spaces
	5 The Euler-Poincaré Characteristic
	6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism
	7 Simplicial Spaces
	8 Simplicial Homology
Chapter VI Functors of Complexes
	1 Modules
	2 Additive Functors
	3 Derived Functors
	4 Univers al Coefficient Formula
	5 Tensor and Torsion Products
	6 Hom and Ext
	7 Singular Homology and Cohomology with General Coefficient Groups
	8 Tensorproduct and Bilinearity
	9 Tensorproduct of Complexes. Künneth Formula
	10 Hom of Complexes. Homotopy Classification of Chain Maps
	11 Acyclic Models
	12 The Eilenberg-Zilber Theorem. Künneth Formulas for Spaces
Chapter VII Products
	1 The Scalar Product
	2 The Exterior Homology Product
	3 The Interior Homology Product (Pontrjagin Product)
	4 Intersection Numbers in R^n
	5 The Fixed Point Index
	6 The Lefschetz-Hopf Fixed Point Theorem
	7 The Exterior Cohomology Product
	8 The Interior Cohomology Product (\cup-Product)
	9 \cup-Products in Projective Spaces. Hopf Maps and Hopf Invariant
	10 Hopf Algebras
	11 The Cohomology Slant Product
	12 The Cap-Product (\cap-Product)
	13 The Homology Slant Product, and the Pontrjagin Slant Product
Chapter VIII Manifolds
	1 Elementary Properties of Manifolds
	2 The Orientation Bundle of a Manifold
	3 Homology of Dimensions > n in n-Manifolds
	4 Fundamental Class and Degree
	5 Limits
	6 Cech Cohomology of Locally Compact Subsets of R^n
	7 Poincaré-Lefschetz Duality
	8 Examples, Applications
	9 Duality in a-Manifolds
	10 Transfer
	11 Thom Class, Thom Isomorphism
	12 The Gysin Sequence. Examples
	13 Intersection of Homology Classes
Appendix Kan- and Cech-Extensions of Functors
	1 Limits of Functors
	2 Polyhedrons under a Space, and Partitions of Unity
	3 Extending Functors from Polyhedrons to More General Spaces
Bibliography
Subject Index