Lectures on Algebraic Topology

Contents
Singular homology
Introduction: singular simplices and chains
Homology
Categories, functors, and natural transformations
Categorical language
Homotopy, star-shaped regions
Homotopy invariance of homology
Homology cross product
Relative homology
Homology long exact sequence
Excision and applications
Eilenberg-Steenrod axioms and the locality principle
Subdivision
Proof of the locality principle
Computational methods
CW complexes I
CW complexes II
Homology of CW complexes
Real projective space
Euler characteristic and homology approximation
Coefficients
Tensor product
Tensor and Tor
Fundamental theorem of homological algebra
Hom and Lim
Universal coefficient theorem
Künneth and Eilenberg-Zilber
Cohomology and duality
Coproducts, cohomology
Ext and UCT
Products in cohomology
Cup product, continued
Surfaces and nondegenerate symmetric bilinear forms
Local coefficients and orientations
Proof of the orientation theorem
A plethora of products
Cap product and Cech cohomology
Cech cohomology as a cohomology theory
Fully relative cap product
Poincaré duality
Applications
Basic homotopy theory
Limits, colimits, and adjunctions
Cartesian closure and compactly generated spaces
Basepoints and the homotopy category
Fiber bundles
Fibrations, fundamental groupoid
Cofibrations
Cofibration sequences and co-exactness
Weak equivalences and Whitehead's theorems
Homotopy long exact sequence and homotopy fibers
The homotopy theory of CW complexes
Serre fibrations and relative lifting
Connectivity and approximation
The Postnikov tower
Hurewicz, Eilenberg, Mac Lane, and Whitehead
Representability of cohomology
Obstruction theory
Vector bundles and principal bundles
Vector bundles
Principal bundles, associated bundles
G-CW complexes and the I-invariance of BunG
The classifying space of a group
Simplicial sets and classifying spaces
The Cech category and classifying maps
Spectral sequences and Serre classes
Why spectral sequences?
Spectral sequence of a filtered complex
Serre spectral sequence
Exact couples
Gysin sequence, edge homomorphisms, and transgression
Serre exact sequence and the Hurewicz theorem
Double complexes and the Dress spectral sequence
Cohomological spectral sequences
Serre classes
Mod C Hurewicz and Whitehead theorems
Freudenthal, James, and Bousfield
Characteristic classes, Steenrod operations, and cobordism
Chern classes, Stiefel-Whitney classes, and the Leray-Hirsch theorem
H*(BU(n)) and the splitting principle
Thom class and Whitney sum formula
Closing the Chern circle, and Pontryagin classes
Steenrod operations
Cobordism
Hopf algebras
Applications of cobordism
Bibliography
Index