##
**Algebraic topology.**
*(English)*
Zbl 1044.55001

Cambridge: Cambridge University Press (ISBN 0-521-79540-0/pbk). xii, 544 p. (2002).

Without any doubt, algebraic topology must be seen as one of the most eminent achievements in modern mathematics. Especially in the second half of the 20th century, this fairly young branch of pure mathematics has undergone tremendous developments in various directions, thereby penetrating (and even revolutionizing) almost all other disciplines of contemporary mathematics and mathematical physics. Accordingly, algebraic topology has become an indispensible, utmost important component of teaching pure mathematics at the graduate university level, at least so with regard to its basic concepts, methods, results, and applications in analysis, geometry, algebra and modern mathematical physics.

Certainly, there are numerous excellent textbooks on algebraic topology that have appeared over the period of the past forty years. Apart from the very standard basics of algebraic topology, those books differ from each other in many regards, depending on the respective author’s individual viewpoint, objective, and aspiration for disciplinary topicality.

However, most of the classic standard texts on the subject are now more than thirty years old, while our understanding of the most important principles and applications of algebraic topology has undergone significant changes over time.

The book under review reflects the author’s efforts to take these developments into account, especially with a view towards their pedagogical consequences, and to throw a bridge between classical and modern teaching of the foundations of algebraic topology.

To begin with, he contrives to reach this goal in a masterly manner. Conceptually the general viewpoint is still quite classical in spirit, yet with a much stronger and reasonable emphasis on the topology of CAN complexes, and the flavor of the entire text is intentionally much more geometrical than algebraic-functorial. In this vein, the four main chapters of the book provide the basic core material of algebraic topology in its two main streams, homotopy theory and homology theory, together with an ample selection of additional, more advanced and optional topics, accounting for nearly half of the book altogether, and offering plenty of basic background material for further studies.

As for the precise contents, the book is subdivided into five chapters.

Chapter 0 is entitled “Some Underlying Geometric Notions”. Being thought of as a user-friendly starter, this chapter introduces some of the very basic geometric concepts and constructions that are crucial for both the homological and homotopical sides of algebraic topology. The reader encounters here, above all, elementary homotopy theory, construction principles for topological spaces, and the first properties of cell complexes.

Chapter 1 comes with the title “The Fundamental Group” and provides most of the standard material on covering spaces, Poincaré groups of topological spaces, the topology of surfaces, and the Van Kampen theorem. This is enriched by two additional topics concerning graphs and free groups as well as \(K(G,1)\) spaces.

Chapter 2 turns to homology theory and discusses, in three main sections, simplicial and singular homology, cellular homology, their geometric applications, and then finally the axiomatic-functorial approach to general homology theory. The optional, further-going topics added to this chapter include the comparison of the first homology group and the fundamental group of a topological space, some of the standard applications such as the theorem of Hopf on division algebras and the Borsuk-Ulam theorem, and the method of simplicial approximation in algebraic topology.

Chapter 3 continues the homological part of the subject by explaining cohomology and the related duality theory for topological spaces. Cohomology groups, cohomology operations, and the classical duality theorems form the core of this chapter, which is then enhanced by eight additional, more specific topics within this context, including H-spaces and Hopf algebras, Bockstein operators, the cohomology of \(\text{SO}(n)\), transfer homomorphisms, and cohomology with local coefficients.

Chapter 4 returns to the concept of homotopy and is devoted to what is usually called higher homotopy theory. Apart from the basics on higher homotopy groups, including Whitehead’s theorem, the Hurewicz theorem, cellular approximation, CW approximation, and the excision properties of higher homotopy groups, the author also discusses the homotopy of fiber bundles, stable homotopy theory, the homotopy construction of cohomology, Moore-Postnikov towers, and the idea of obstruction theory. This somewhat more advanced chapter comes with even twelve additional topics in the sequel, amongst which are such specialties like the cohomology of fiber bundles, the Eckmann-Hilton duality, loop spaces, the Dold-Thom theorem, and the various Steenrod cohomology operations.

The book ends with an appendix reviewing the general topology of cell complexes and the peculiarities of the compact-open topology.

As a whole, the text is arranged in such a way that both the learning student and the teacher in the field have various choices how to advance, depending on their special likings or targets. In fact, this textbook covers a rather broad spectrum of the overall scheme of algebraic topology, in its geometric aspects, and the truly unusual abundance of instructive examples and complementing exercises is absolutely unique of such a kind. Also, the distinctly circumspect, methodologically inductive, intuitive, descriptively elucidating and very detailed style of writing give evidence to the fact that the author’s first priorities are exactly what students need when working with such a textbook, namely clarity, readability, steady motivation, guided inspiration, increasing demand, and as much self-containedness of the exposition as possible. No doubt, a very devoted and experienced teacher has been at work here, very much so to the benefit of beginners in the field of algebraic topology, instructors, and interested readers in general. With this text as a profound background, the reader will be perfectly able to continue his studies by means of the more advanced, more topical, more algebraically flavored, and then also more concise treatises of algebraic topology pointed out in the bibliography. As for further reading, also the recent textbooks by J. P. May [A concise course in algebraic topology, Chicago Lectures in Mathematics, Univ. Press, Chicago (1999; Zbl 0923.55001)], T. tom Dieck [Topologie, 2nd edition, Berlin-New York, W. de Gruyter Verlag (2000; Zbl 0966.55001)], and R. Stöcker and H. Zieschang [Algebraische Topologie (B. G. Teubner Verlag, Stuttgart) (1994; Zbl 0823.55001)] should be seriously taken into consideration.

Finally, as the author points out, his excellent introduction to algebraic topology will remain available online in electronic form. The web address is http://www.math.cornell.edu/\(^\sim\)hatcher, and one can also find here parts of some other (forthcoming) books of the authors, which complement the present introductory text in various directions.

Certainly, there are numerous excellent textbooks on algebraic topology that have appeared over the period of the past forty years. Apart from the very standard basics of algebraic topology, those books differ from each other in many regards, depending on the respective author’s individual viewpoint, objective, and aspiration for disciplinary topicality.

However, most of the classic standard texts on the subject are now more than thirty years old, while our understanding of the most important principles and applications of algebraic topology has undergone significant changes over time.

The book under review reflects the author’s efforts to take these developments into account, especially with a view towards their pedagogical consequences, and to throw a bridge between classical and modern teaching of the foundations of algebraic topology.

To begin with, he contrives to reach this goal in a masterly manner. Conceptually the general viewpoint is still quite classical in spirit, yet with a much stronger and reasonable emphasis on the topology of CAN complexes, and the flavor of the entire text is intentionally much more geometrical than algebraic-functorial. In this vein, the four main chapters of the book provide the basic core material of algebraic topology in its two main streams, homotopy theory and homology theory, together with an ample selection of additional, more advanced and optional topics, accounting for nearly half of the book altogether, and offering plenty of basic background material for further studies.

As for the precise contents, the book is subdivided into five chapters.

Chapter 0 is entitled “Some Underlying Geometric Notions”. Being thought of as a user-friendly starter, this chapter introduces some of the very basic geometric concepts and constructions that are crucial for both the homological and homotopical sides of algebraic topology. The reader encounters here, above all, elementary homotopy theory, construction principles for topological spaces, and the first properties of cell complexes.

Chapter 1 comes with the title “The Fundamental Group” and provides most of the standard material on covering spaces, Poincaré groups of topological spaces, the topology of surfaces, and the Van Kampen theorem. This is enriched by two additional topics concerning graphs and free groups as well as \(K(G,1)\) spaces.

Chapter 2 turns to homology theory and discusses, in three main sections, simplicial and singular homology, cellular homology, their geometric applications, and then finally the axiomatic-functorial approach to general homology theory. The optional, further-going topics added to this chapter include the comparison of the first homology group and the fundamental group of a topological space, some of the standard applications such as the theorem of Hopf on division algebras and the Borsuk-Ulam theorem, and the method of simplicial approximation in algebraic topology.

Chapter 3 continues the homological part of the subject by explaining cohomology and the related duality theory for topological spaces. Cohomology groups, cohomology operations, and the classical duality theorems form the core of this chapter, which is then enhanced by eight additional, more specific topics within this context, including H-spaces and Hopf algebras, Bockstein operators, the cohomology of \(\text{SO}(n)\), transfer homomorphisms, and cohomology with local coefficients.

Chapter 4 returns to the concept of homotopy and is devoted to what is usually called higher homotopy theory. Apart from the basics on higher homotopy groups, including Whitehead’s theorem, the Hurewicz theorem, cellular approximation, CW approximation, and the excision properties of higher homotopy groups, the author also discusses the homotopy of fiber bundles, stable homotopy theory, the homotopy construction of cohomology, Moore-Postnikov towers, and the idea of obstruction theory. This somewhat more advanced chapter comes with even twelve additional topics in the sequel, amongst which are such specialties like the cohomology of fiber bundles, the Eckmann-Hilton duality, loop spaces, the Dold-Thom theorem, and the various Steenrod cohomology operations.

The book ends with an appendix reviewing the general topology of cell complexes and the peculiarities of the compact-open topology.

As a whole, the text is arranged in such a way that both the learning student and the teacher in the field have various choices how to advance, depending on their special likings or targets. In fact, this textbook covers a rather broad spectrum of the overall scheme of algebraic topology, in its geometric aspects, and the truly unusual abundance of instructive examples and complementing exercises is absolutely unique of such a kind. Also, the distinctly circumspect, methodologically inductive, intuitive, descriptively elucidating and very detailed style of writing give evidence to the fact that the author’s first priorities are exactly what students need when working with such a textbook, namely clarity, readability, steady motivation, guided inspiration, increasing demand, and as much self-containedness of the exposition as possible. No doubt, a very devoted and experienced teacher has been at work here, very much so to the benefit of beginners in the field of algebraic topology, instructors, and interested readers in general. With this text as a profound background, the reader will be perfectly able to continue his studies by means of the more advanced, more topical, more algebraically flavored, and then also more concise treatises of algebraic topology pointed out in the bibliography. As for further reading, also the recent textbooks by J. P. May [A concise course in algebraic topology, Chicago Lectures in Mathematics, Univ. Press, Chicago (1999; Zbl 0923.55001)], T. tom Dieck [Topologie, 2nd edition, Berlin-New York, W. de Gruyter Verlag (2000; Zbl 0966.55001)], and R. Stöcker and H. Zieschang [Algebraische Topologie (B. G. Teubner Verlag, Stuttgart) (1994; Zbl 0823.55001)] should be seriously taken into consideration.

Finally, as the author points out, his excellent introduction to algebraic topology will remain available online in electronic form. The web address is http://www.math.cornell.edu/\(^\sim\)hatcher, and one can also find here parts of some other (forthcoming) books of the authors, which complement the present introductory text in various directions.

Reviewer: Werner Kleinert (Berlin)

### MathOverflow Questions:

How do I show that any finite-dimensional (absolute) CW-complex \(X\) is locally contractible?Nerve theorem for locally infinite covers by subcomplexes

### MSC:

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

55Nxx | Homology and cohomology theories in algebraic topology |

55Pxx | Homotopy theory |

55Qxx | Homotopy groups |