Lecture notes in algebraic topology / James F. Davis, Paul Kirk.

Includes bibliographical references (p. 359-361) and index.

Chapters
Chapter 1. Chain complexes, homology, and cohomology
Chapter 2. Homological algebra
Chapter 3. Products
Chapter 4. Fiber bundles
Chapter 5. Homology with local coefficients
Chapter 6. Fibrations, cofibrations and homotopy groups
Chapter 7. Obstruction theory and Eilenberg-MacLane spaces
Chapter 8. Bordism, spectra, and generalized homology
Chapter 9. Spectral sequences
Chapter 10. Further applications of spectral sequences
Chapter 11. Simple-homotopy theory

Volume: 35; 2001; 367 pp
MSC: Primary 55; 57;
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems.

To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.

Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.

The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic K
-theory and the s-cobordism theorem.

A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars.

The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.

Readership
Graduate students and research mathematicians interested in geometric topology and homotopy theory.