Textbook in PDF format

Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics. These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT. The style is engaging and student-friendly, but precise. Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory. Preface 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 Postnikov towers 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