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Marco Manetti
Publication Date: 
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[Reviewed by
Mark Hunacek
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This is a textbook in point-set and introductory algebraic topology, intended for an audience of upper-level undergraduates. The book divides naturally into two parts of roughly equal length. As explained in more detail below, chapters 2 through 8 cover the basic content of a one-semester course in point-set topology, and chapters 10 through 15 introduce algebraic topology. The two chapters not listed above, 1 and 9, are prefatory in nature and focus on examples and intuition rather than rigorous proofs; chapter 1 is designed to introduce the material of chapters 2 through 8, and chapter 9 (“Intermezzo”) provides a quick introduction to what algebraic topology is about, before settling in for the details that follow in the next six chapters.

Here’s a bit more detail about what is covered: the first part of the book begins with a chapter on set theory (the usual countability/Zorn stuff) and then proceeds through a discussion of topological spaces (metric spaces being introduced later in the chapter as a major example), connectedness and compactness, quotient spaces, first and second countable spaces, product spaces and Tychonoff’s theorem, complete metric spaces and the Baire category theorem, normal spaces and the separation axioms, and other topics (the compact-open topology, paracompactness, function spaces and Arzelà-Ascoli, etc.) Urysohn’s Lemma and the Tietze Extension theorem are presented as do-it-yourself exercises, broken up into more or less bite-sized bits.

The second part of the book begins by defining homotopy and the fundamental group, then moving into covering spaces and applications, including ones to free groups, the fundamental theorem of algebra, and the hairy ball theorem (that I learned as “you can’t comb a coconut”). An introduction to the language of categories and functors is also given. More sophisticated algebraic topology, such as homology and cohomology, is not covered (nor should it be, at this level); the nth homotopy group is mentioned very briefly in passing, but nothing is done with it. Of course, not many undergraduates see these topics nowadays, and so this part of the book will probably not find much use at the undergraduate level; I suspect that’s just as well, since I’m not sure most undergraduates would find this part of the text as easy as the first part.

Readers who are familiar with Munkres’ Topology will recognize a certain similarity in the structure of both books, but there are some differences as well. Some of these differences are in content coverage: Munkres discusses some fairly sophisticated results not mentioned in this book (examples: the Nagata-Smirnov metrization theorem and the Jordan Curve theorem). Other differences concern the overall level of difficulty of the texts. Manetti seems to spend more time trying to motivate things for the student than does Munkres, whose text does not, for example, have any chapters that are comparable to Manetti’s chapters 1 and 9. Overall I would say that Manetti’s text is somewhat more reader-friendly than is Munkres’, and for that reason more likely to be successful as a text at a typical university.

Besides the two introductory chapters, there are other things about this text that make it reader-friendly. The author writes clearly and provides lots of examples, and there are also a good number of exercises, solutions to a relatively small number of which are available in a 25-page concluding chapter. The exercises range from routine to quite challenging. (I was amused by the notation that the author uses in the exercise sets: those exercises that have solutions are denoted by a heart symbol, perhaps because the students love seeing solutions; those exercises that the author views as more difficult are marked by a coffee-cup symbol).

I also liked the emphasis on matrix groups as examples of topological spaces: Manetti has a section on topological groups in chapter 4 (“Connectedness and Compactness”) that contains proofs of a number of standard but (to my mind, anyway) very interesting facts about the compactness and connectedness of the general linear group and some of its famous subgroups, as well as general theorems about topological groups. Then later, in the second part of the book, there is a section discussing topological aspects of SO(3) in more detail.

On the other hand, there were some things about the book that I didn’t like so much, starting with (in common with Munkres) the introduction of topological spaces before metric spaces. Maybe it’s just because I learned it that way, but I have always felt that doing metric spaces first offers significant pedagogical advantages. Dealing with metric spaces is a relatively easy step from the student’s experiences with calculus or analysis, and topological spaces, in turn, are much easier to motivate once the student has spent time working with metric spaces. (This is the approach taken in Conway’s A Course in Point-Set Topology, and I think it is a good one.)

As for the contents of the text, while all the standard topics that I would want to cover in an introductory point-set topology course are mentioned in part I, some are done so in less detail than I would prefer. The Banach contraction mapping theorem, for example, appears only as an exercise, with none of the interesting applications of that theorem mentioned. It seems to me, though, that using that theorem to prove an existence-uniqueness ODE theorem is not terribly hard, and gives a splendid example of the power of topology in solving problems in analysis. Likewise, although the Baire category theorem is proved, no real indication of its usefulness is provided.

I also thought the index could have been improved. Some important items like paracompactness (defined on page 134 of the text) and the Fundamental Theorem of Algebra (given two different proofs in the text) are not mentioned in the index at all, and when an item is mentioned, there are sometimes inadequate references: for example, there is only one entry for “Möbius strip”, and that’s to page 5, where it is introduced informally in chapter 1; there is no reference to the page (91) where it is defined precisely as a quotient space. A notation index would also have been helpful.

And speaking of notation, I am strenuously opposed to denoting the open interval (a, b) by ]a, b[. This backwards-bracket notation seems unpleasant to me, particularly when used in conjunction with regular brackets to denote half-open intervals: the notation (0 ,1] seems much more intuitive and easy to read than ]0,1].

Moreover, although, as I said earlier, the writing style is generally clear and easy to understand, there are occasional passages that remind the reader that this book is a translation from the Italian. (Example: “The object of concern in this section is the topological group SO(3,R), the importance of whom, especially in solid-state mechanics and particle physics, cannot go amiss.”) I did not, however, see any examples where the translation made it difficult to tell what thought the author was trying to convey, however.

However, these are all relatively minor quibbles, and are easily outweighed by the good features of this book. Overall, this is a solid entry in the undergraduate-topology textbook market, and anybody teaching a course in the subject should take a look at it.

Mark Hunacek ( teaches mathematics at Iowa State University.