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Topology and groupoids. 3rd revised, updated and extended ed. (English) Zbl 1093.55001

Bangor: Ronald Brown (ISBN 1-4196-2722-8/pbk). xxiv, 512 p. (2006).
This is the third edition of a book that originally appeared in 1968 with the title “Elements of Modern Topology” (1968; Zbl 0159.52201). The second edition appeared in 1988 with the title “Topology: A Geometric Account of General Topology, Homotopy types and the Fundamental Groupoid” (1988; Zbl 0655.55001). Of the three titles, the current one being “Topology and Groupoids”, that of the second edition, perhaps because it is the longest, gives the best description of the book’s contents. Rather than reproducing the table of contents chapter by chapter, it perhaps makes more sense to give an overview of the “narrative structure” of the book. The material in the book is in a large sense determined by the author’s avowed intention to encourage the study of the fundamental groupoid of a topological space, rather than the fundamental group. The objects of the fundamental groupoid \(\pi X\) of a space \(X\) are the equivalence classes \(\pi X(x,y)\) of paths from \(x\) to \(y\) in \(X\) under a relation of homotopy. By the conclusion of the book the author has proved the Van Kampen theorem for adjunction spaces and discussed covering spaces and groupoids, as well as orbit spaces and orbit groupoids. The use of groupoids rather than groups allows for a more categorical approach than would otherwise be possible.
The book begins with a review of basic point set topology, particular attention is paid to connectedness and local connectedness, path connectedness, compactness and Hausdorff spaces as one would expect from the author’s geometric aims. The role of connectivity in distinguishing homeomorphism types of spaces is introduced. This serves to justify the need for better machinery while showing the fundamental role of connectivity in geometric topology.
There is a detailed discussion of identification spaces and adjunction spaces – with the idea that these spaces are the ones for which the categorical machinery can be set up in a way that is reflected by the fundamental groupoid. There is an interesting discussion of the topology of product spaces. If one is studying mapping spaces it is desirable that the set map \(Z^{X \times Y} \rightarrow (Z^{Y})^{X}\) be a continuous bijection. This is not always the case, fortunately this is the case for identification spaces that are the sum of compact Hausdorff spaces.
With the topological preliminaries out of the way the author can turn to the definition of the fundamental groupoid and the use of pushouts and cofibrations for the construction of spaces.
The book contains numerous exercises and each chapter concludes with extensive historical notes. The author says that he wrote the book in part to understand the original arguments in the literature. This means that the reader is given complete proofs, without being referred to exercises to complete them. Beginning students of homotopy theory will likely find the book’s level of detail to be at one and the same time both off-putting and helpful. As an introduction to algebraic topology many will find the absence of homology and cohomology a disadvantage. On the other hand this is entirely consistent with the author’s emphasis on the fundamental groupoid. A minor cavil is the use of \(XA\) for \(X \cap A\) in the groupoid \(\pi XA\) which obscures the symmetrical role of \(X\) and \(A\) and is quite non-standard.

MSC:

55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
54-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
57M05 Fundamental group, presentations, free differential calculus
18F99 Categories in geometry and topology
57M10 Covering spaces and low-dimensional topology
54B30 Categorical methods in general topology
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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