MR0227979 (37 #3563) Brown, Ronald
Elements of modern topology. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1968 xvi+351 pp.

This new topology text, intended for various levels of students, provides a fresh and modern approach. Its most unusual feature is the abstract treatment of the fundamental group; this treatment is based on the recently-developed theory of groupoids. The book is geared to algebraic topology, especially homotopy theory. While the attitude in early chapters is highly concrete and geometrical, the level of difficulty increases and the latter portions are more abstract and functorial. An important aspect is the inclusion of much of the sort of point-set topology which is useful mainly to the algebraic topologist. Specifically, there is substantial material on identification spaces, cell complexes, joins, and smashed products. These topics give this book a real usefulness to the specialist.

The author indicates in his preface some potential uses for this book as a text. He suggests, in addition to shorter courses, both a two-term beginning undergraduate course in general and algebraic topology and a two-term master's-level course dealing mostly with cell complexes and the fundamental groupoid (including covering spaces). Some instructors may prefer material more oriented toward analysis in the first course, but those who wish to go directly toward homotopy theory will like this book. They will like its clarity of presentation, its completeness in those areas which it aims to cover, and its very rich supply of exercises.

The first three chapters form a selective introduction to general topology; those topics covered comprise that portion of standard point-set theory which is needed for homotopy theory. Chapter 1 studies elementary topological properties of the real line. Chapter 2 contains basic material on topological spaces, with emphasis on metric spaces, while Chapter 3 deals with compactness and connectedness.

Chapters 4 and 5 include additional point-set results which are needed for homotopy theory. Chapter 4 is on identification spaces and cell complexes, and Chapter 5 contains more specialized topics, such as projective spaces (with their cell structures), joins, and smashes.

The final four chapters are more specialized and comprise the algebraic topology section of the book. They center mostly, but not exclusively, on groupoids, especially the fundamental groupoid of a space. It is in this portion of the book that the theme clearly emerges. This theme is the close interaction of topology and algebra which appears in the relation of a space to its fundamental groupoid; this interaction is stressed not only for its sake, but as an elegant example of the modern point of view in mathematics.

Chapter 6 is devoted specifically to the fundamental groupoid, with emphasis on its algebraic and categorical structure. The author returns in Chapter 7 to homotopy theory with a study of the homotopy extension property; a proof of the cellular approximation theorem is included. Computations, based on the Van Kampen theorem, appear in Chapter 8. These computations are handled quite abstractly; the Van Kampen theorem itself is stated in terms of a pushout diagram of groupoids. Chapter 9 presents the theory of covering spaces, covering groupoids, and their relationship.

There is included a glossary on basic, and an appendix on not-so-basic, results needed from set theory.

Reviewed by R. E. Mosher

MR0984598 (90k:54001) Brown, Ronald(4-NWAL)
Topology. A geometric account of general topology, homotopy types and the fundamental groupoid. Second edition. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1988. xviii + 460

This book has been out of print since the first edition ended its print run in 1972. It is nice to have it back in a new edition. Although a number of changes have been made to it, the book is still intended as a text for a two-semester course in topology and algebraic topology at the advanced undergraduate or beginning graduate level. The general direction of the book is toward homotopy theory with a geometric point of view. This book would provide a more than adequate background for a standard algebraic topology course that begins with homology theory. The first three chapters deal with the standard topics in point set topology.

Chapter 1 features the real line, with its usual topology. Included there is a nice treatment of the Cantor set. Chapter 2 deals with general topological spaces, maps between them, subspaces, product spaces, and metric spaces. It is a fairly long chapter with almost 100 exercises included. The third chapter covers compactness, connected spaces and separation axioms. Some of the deeper results, like the Tietze extension theorem and the Tikhonov theorem are stated and discussed here, but not proved. Other topics discussed include proper maps and one-point compactifications.

The next two chapters mark a departure from standard texts. Chapter 4 covers identification spaces, adjunction spaces and cell complexes in detail (40 pages). A number of specific examples are described at length in Chapter 5. They include the CW decomposition of projective spaces, simplicial complexes, joins,  smash products, and function spaces. Here the author uses the idea of a "$k$-space" to topologize function spaces and products so that the appropriate exponential laws hold. This idea goes back to his Oxford thesis. There and in subsequent papers he showed how "$k$-spaces" provide a convenient category of topological spaces. Today "$k$-spaces" are better known as compactly generated spaces.

The last four chapters deal with groupoids. Chapter 6 covers the construction of the fundamental groupoid, basic properties of groupoids, morphisms, and functors of groupoids. It is then shown that the fundamental groupoid takes a pushout of spaces to a pushout of groupoids. This leads to a proof of the van Kampen theorem in a later chapter. Chapter 7 starts off with cofibrations and track groups. The new notion of a fibration of groupoids is introduced. A gluing theorem for homotopy equivalence is proved as is the cellular approximation theorem. In Chapter 8, the van Kampen theorem and the Jordan curve theorem are proved. Both proofs make essential use of groupoids. The last chapter, 9, deals with covering spaces. They too are treated in terms of groupoids.  Some applications of covering spaces to group theory are included, e.g. the Nielsen-Schreier theorem, the Kurosh theorem, and Grushko's theorem.

One of the nicest features of this book is the joy and enthusiasm that pervade it. A quote from the introduction illustrates this best: "As to the practicalities (of doing mathematical research), I remember thinking to myself after a long session with Michael: `If Michael Barratt can try out one damn fool thing after another, why can't I? This has seemed a reasonable way of proceeding ever since. What is not so clear is why the really foolish projects (such as higher homotopy groupoids, based on flimsy evidence and counter to current traditions) have turned out the most fun."

Reviewed by Charles A. McGibbon