##
**Lecture notes in algebraic topology.**
*(English)*
Zbl 1018.55001

Graduate Studies in Mathematics. 35. Providence, RI: AMS, American Mathematical Society. xvi, 367 p. (2001).

This book is based on the lecture notes of the authors’ courses at Indiana University. It targets readers with a previous exposure to basic algebraic topology notions like the fundamental group, singular (co)homology, and CW complexes. By its declared purpose, it aims to cover “what every young topologist should know” in a self-contained, easy to read and reasonably short volume.

In order to achieve their goal, the authors decided to separate the technical difficulties from the underlying ideas. Their main technical tool is a spectral sequence converging to an additive homology theory (e.g., singular homology, K-theory, framed bordism) of the total space of a fibration, obtained by combining the Leray-Serre and the Atiyah-Hirzebruch spectral sequences.

The first four chapters cover the basic constructions of singular and cellular (co)homology, homological algebra, cohomology products and fiber bundles. Chapter 5 introduces homology with local coefficients. Chapter 6 is one of the main parts of the book. Here the authors introduce compactly generated spaces, (co)fibrations, loop spaces and homotopy groups. The Hurewicz comparison theorem between homotopy and homology groups is stated here but proved later using spectral sequences. Chapter 7 covers obstruction theory and Eilenberg-MacLane spaces. Chapter 8 treats bordism, classifying spaces, spectra and generalized homology theories. Chapter 9 forms the technical core of the book, through the Leray-Serre-Atiyah-Hirzebruch spectral sequence. It is perhaps unfortunate that the construction of this spectral sequence is not explicitly given, however the reader is referred to G. W. Whitehead [Elements of homotopy theory; Graduate Texts in Mathematics 61, Springer-Verlag, Berlin-Heidelberg-New York (1978; Zbl 0406.55001)]. Another good reference for spectral sequences is R. Bott and L. Tu [Differential forms in algebraic topology; Graduate Texts in Mathematics 82, Springer-Verlag, New York-Heidelberg-Berlin (1982; Zbl 0496.55001)]. Here and in Chapter 10, this spectral sequence gives effortless proofs for the Hurewicz theorem, the Freudenthal suspension theorem and allows several computations of homology and homotopy groups. It also leads to cohomology operations. The book concludes with a chapter on simple-homotopy theory, including Reidemeister torsion, algebraic \(K\)-theory and \(s\)-cobordism.

The book contains more than two hundred exercises scattered throughout the text (as part of the authors’ strategy to avoid technicalities, sometimes proofs of theorems are proposed as exercises). In addition, each chapter ends with several “Projects”; when the book is used as a textbook, the projects could serve as topics for students’ presentations.

The book is very carefully written. The style is alert, sometimes informal, and highly readable. Through their use of modern notation the authors manage to avoid intimidating commutative diagrams without compromising on rigor. In our view, this book represents a valuable addition to the literature which should be useful both to students and as a reference for working mathematicians.

In order to achieve their goal, the authors decided to separate the technical difficulties from the underlying ideas. Their main technical tool is a spectral sequence converging to an additive homology theory (e.g., singular homology, K-theory, framed bordism) of the total space of a fibration, obtained by combining the Leray-Serre and the Atiyah-Hirzebruch spectral sequences.

The first four chapters cover the basic constructions of singular and cellular (co)homology, homological algebra, cohomology products and fiber bundles. Chapter 5 introduces homology with local coefficients. Chapter 6 is one of the main parts of the book. Here the authors introduce compactly generated spaces, (co)fibrations, loop spaces and homotopy groups. The Hurewicz comparison theorem between homotopy and homology groups is stated here but proved later using spectral sequences. Chapter 7 covers obstruction theory and Eilenberg-MacLane spaces. Chapter 8 treats bordism, classifying spaces, spectra and generalized homology theories. Chapter 9 forms the technical core of the book, through the Leray-Serre-Atiyah-Hirzebruch spectral sequence. It is perhaps unfortunate that the construction of this spectral sequence is not explicitly given, however the reader is referred to G. W. Whitehead [Elements of homotopy theory; Graduate Texts in Mathematics 61, Springer-Verlag, Berlin-Heidelberg-New York (1978; Zbl 0406.55001)]. Another good reference for spectral sequences is R. Bott and L. Tu [Differential forms in algebraic topology; Graduate Texts in Mathematics 82, Springer-Verlag, New York-Heidelberg-Berlin (1982; Zbl 0496.55001)]. Here and in Chapter 10, this spectral sequence gives effortless proofs for the Hurewicz theorem, the Freudenthal suspension theorem and allows several computations of homology and homotopy groups. It also leads to cohomology operations. The book concludes with a chapter on simple-homotopy theory, including Reidemeister torsion, algebraic \(K\)-theory and \(s\)-cobordism.

The book contains more than two hundred exercises scattered throughout the text (as part of the authors’ strategy to avoid technicalities, sometimes proofs of theorems are proposed as exercises). In addition, each chapter ends with several “Projects”; when the book is used as a textbook, the projects could serve as topics for students’ presentations.

The book is very carefully written. The style is alert, sometimes informal, and highly readable. Through their use of modern notation the authors manage to avoid intimidating commutative diagrams without compromising on rigor. In our view, this book represents a valuable addition to the literature which should be useful both to students and as a reference for working mathematicians.

Reviewer: Sergiu Moroianu (Toulouse)