Algebraic Topology

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The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.

The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides. The readership of the n-Lab is not restricted to Algebraic Topologists of course, but the above list of notes would be useful to have readily available. Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and a full exposition of Steenrod squares and powers.

This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. To cite examples, I find chapter 0 unreadable, especially the bits about CW complexes (I feel that the proofs in chapter 0 are at best incomplete but I may be grossly mistaken) and also example 0. The electronic version has narrower margins than the print version for a better reading experience on portable electronic devices. This is the first place I've found explanations (that I understand) of things like Mayer-Vietoris sequences of homotopy groups, homotopy pushout and pullback squares etc.

Along with the basic material on fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory, the book includes many optional topics for which elementary expositions are hard to find. Homologie singulière des éspaces fibrés has as clear and economical an account of spectral sequences as I've seen anywhere. Algebraic topoligy books that emphasize geometrical intuition usually have only a modest technical reach. Revisions and Additions: I have made a number of small changes in the text itself in the years since the book was first published.This book deals with a hard subject, but every effort has been made to explain and motivate the ideas involved before they are dealt with rigorously.