A concise course in algebraic topology.

*(English)*Zbl 0923.55001
Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ix, 243 p. (1999).

This new book on algebraic topology, written by one of the most experienced researchers and teachers in the field, grew out of courses that the author has taught repeatedly, since around 1970, at the renowned University of Chicago. According to the specific first year graduate program in mathematics there, he has been trying, over the past decades, to squeeze a profound introduction to pure algebraic topology into a single one quarter course of lectures on this subject. Due to the fact that a large number of graduate students at the University of Chicago traditionally tend to specialize in topology, both algebraic and geometric, it appears as an additional requirement that such a course should serve two main purposes simultaneously: providing the basic material on algebraic topology that is needed in various branches of mathematics, on the one hand, and laying the foundations for later research in the field, on the other hand. Moreover, algebraic topology has undergone subtle but significant internal changes, since the appearance of most standard textbooks on this subject, but those changes have barely been noticed by the non-experts.

Thus, apart from his afore-mentioned twofold pedagogical task in providing a concise introduction to algebraic topology, the author has also tried to offer an approach that reflects some of these new developments in the field. In this vein, his approach focuses conceptually on the algebraic homotopical methods of constructing generalized (co-)homology theories, which indeed have become increasingly important in contemporary mathematics, but which – in contrast to this very fact – have not yet found an adequate systematic, rigorous exposition in the textbook literature, at any level. In a body, these notes reflect the author’s effort to organize a modern and concise introduction to algebraic topology in a way that caters to all these ambitious goals.

The outcome is an excellent, very concise and pretentious, fairly sophisticated and deep-going, but completely self-contained and masterly arranged new textbook, which undoubtedly will become one of the modern standard references in the field.

As to the contents, the text consists of twenty-five chapters (sic !) which are (almost unbelievably) compressed on 230 pages only. These chapters are arranged as follows: 1. The fundamental group and some of its applications; 2. Categorical language and the van Kampen theorem; 3. Covering spaces; 4. Graphs; 5. Compactly generated spaces; 6. Cofibrations; 7. Fibrations; 8. Based cofiber and fiber sequences; 9. Higher homotopy groups; 10. CW-complexes; 11. The homotopy excision and suspension theorems; 12. A little homological algebra; 13. Axiomatic and cellular homology theory; 14. Derivations of properties from the axioms; 15. The Hurewicz and uniqueness theorems; 16. Singular homology theory; 17. Some more homological algebra; 18. Axiomatic and cellular cohomology theory; 19. Derivations of properties from the axioms; 20. The Poincaré duality theorem; 21. The index of manifolds and manifolds with boundary; 22. Homology, cohomology, and Eilenberg-MacLane spaces; 23. Characteristic classes of vector bundles; 24. An introduction to \(K\)-theory; 25. An introduction to cobordism theory.

The last four chapters discuss the material in a slightly more sketchy form. In fact, the advanced material presented here appears as an expanded version of the author’s purposeful digressions in practical teaching, which he uses in order to include both important applications and brief introductions to other central topics in topology. As for the material discussed in the first 21 chapters, that is, for the regular content of the formal introductory course, full proofs are given, and most chapters come with a set of further-leading problems. In view of the fact that the really concise text focuses on the ideas, concepts, methods, and general theorems, the exercises also serve the purpose of developing the computational skills of the reader, which are certainly necessary to assimilate the wealth of condensed information delivered by this comprehensive text.

The book ends with an utmost useful, idiosyncratic guide to the current literature in algebraic topology, basically designed for those readers who are interested in going further in this direction.

In the very instructive preface to his book, the author points out that these notes, although having grown out from a basic course, are not primarily intended as a regular textbook. They might be, in fact, too concise for beginners who want to learn about algebraic topology just by reading, but they positively represent a brilliant textbook for an exceptionally strong graduate course, and they are surely a perfect modern reference book for teachers of the subject. Without any doubt, this book, the text of which evolved from the long-standing experience of a leading researcher and teacher in the field, is a highly valuable enhancement of the standard literature in algebraic topology.

Thus, apart from his afore-mentioned twofold pedagogical task in providing a concise introduction to algebraic topology, the author has also tried to offer an approach that reflects some of these new developments in the field. In this vein, his approach focuses conceptually on the algebraic homotopical methods of constructing generalized (co-)homology theories, which indeed have become increasingly important in contemporary mathematics, but which – in contrast to this very fact – have not yet found an adequate systematic, rigorous exposition in the textbook literature, at any level. In a body, these notes reflect the author’s effort to organize a modern and concise introduction to algebraic topology in a way that caters to all these ambitious goals.

The outcome is an excellent, very concise and pretentious, fairly sophisticated and deep-going, but completely self-contained and masterly arranged new textbook, which undoubtedly will become one of the modern standard references in the field.

As to the contents, the text consists of twenty-five chapters (sic !) which are (almost unbelievably) compressed on 230 pages only. These chapters are arranged as follows: 1. The fundamental group and some of its applications; 2. Categorical language and the van Kampen theorem; 3. Covering spaces; 4. Graphs; 5. Compactly generated spaces; 6. Cofibrations; 7. Fibrations; 8. Based cofiber and fiber sequences; 9. Higher homotopy groups; 10. CW-complexes; 11. The homotopy excision and suspension theorems; 12. A little homological algebra; 13. Axiomatic and cellular homology theory; 14. Derivations of properties from the axioms; 15. The Hurewicz and uniqueness theorems; 16. Singular homology theory; 17. Some more homological algebra; 18. Axiomatic and cellular cohomology theory; 19. Derivations of properties from the axioms; 20. The Poincaré duality theorem; 21. The index of manifolds and manifolds with boundary; 22. Homology, cohomology, and Eilenberg-MacLane spaces; 23. Characteristic classes of vector bundles; 24. An introduction to \(K\)-theory; 25. An introduction to cobordism theory.

The last four chapters discuss the material in a slightly more sketchy form. In fact, the advanced material presented here appears as an expanded version of the author’s purposeful digressions in practical teaching, which he uses in order to include both important applications and brief introductions to other central topics in topology. As for the material discussed in the first 21 chapters, that is, for the regular content of the formal introductory course, full proofs are given, and most chapters come with a set of further-leading problems. In view of the fact that the really concise text focuses on the ideas, concepts, methods, and general theorems, the exercises also serve the purpose of developing the computational skills of the reader, which are certainly necessary to assimilate the wealth of condensed information delivered by this comprehensive text.

The book ends with an utmost useful, idiosyncratic guide to the current literature in algebraic topology, basically designed for those readers who are interested in going further in this direction.

In the very instructive preface to his book, the author points out that these notes, although having grown out from a basic course, are not primarily intended as a regular textbook. They might be, in fact, too concise for beginners who want to learn about algebraic topology just by reading, but they positively represent a brilliant textbook for an exceptionally strong graduate course, and they are surely a perfect modern reference book for teachers of the subject. Without any doubt, this book, the text of which evolved from the long-standing experience of a leading researcher and teacher in the field, is a highly valuable enhancement of the standard literature in algebraic topology.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

55P99 | Homotopy theory |

55N99 | Homology and cohomology theories in algebraic topology |

19L99 | Topological \(K\)-theory |