Most mathematicians have at least a vague understanding of what a manifold is — a topological space which, in a sense which this book of course makes precise, “looks like” a Euclidean space in a neighborhood of each of its points. As the author points out early on, manifolds play a role in nearly every branch of mathematics, and their study should be a part of every graduate student’s education. This book is his attempt to provide that introduction.
Its title notwithstanding, Introduction to Topological Manifolds is, however, more than just a book about manifolds — it is an excellent introduction to both point-set and algebraic topology at the early-graduate level, using manifolds as a primary source of examples and motivation. This approach allows graduate students some exposure to the topic at a relatively early stage of their careers, without having to wait for much more sophisticated and specialized courses in differential or Riemannian geometry. Its prerequisites are modest: aside from the inevitably stated “mathematical maturity” that one would expect from a graduate student, a good knowledge of advanced calculus and some exposure to linear and abstract algebra are necessary; three appendices on set theory, metric spaces and group theory summarize the essentials of these topics (and provide exercises) but a student with no prior exposure to them would probably be at some disadvantage.
After a pleasant and informative introductory chapter in which the author discusses how manifolds appear in a variety of different contexts (including, for example, algebraic geometry, relativity, classical mechanics, and string theory), the book breaks down naturally into four basic parts. The first part, chapters 2 through 4, constitutes a concise introduction to point-set topology. Starting with the definition of a topological space as a generalization of a metric space, the author proceeds to discuss, among other things: continuity, product and quotient spaces (including an introduction to infinite products), topological groups and group actions, compactness, connectedness, and paracompactness. This part of the book, comprising roughly a hundred pages, probably covers most of the material one might expect to find in an introductory topology course (with, of course, an emphasis on manifolds), but some general topology topics that are not necessary for the rest of this text are omitted. Filters and nets do not appear, for example, and Tychonoff’s theorem for infinite products, though mentioned, is not proved (though the finite version is). The Baire Category Theorem is also stated and proved, but some of the standard applications that one might see in an introductory topology course (such as the existence of continuous, non-differentiable functions) are not discussed. In addition, some examples that one might expect to see in a general topology course, such as Möbius strips and Klein bottles, are not discussed here either (although the general subject of quotient spaces is discussed), but they do show up a little later in the text, in chapter 6.
Chapters 5 and 6 discuss CW complexes and the classification of compact surfaces. CW complexes are, roughly speaking, topological spaces obtained by stitching together spaces that are homeomorphic to Euclidean balls; their use here is perhaps the major distinction between the first and second editions of this text — the first edition emphasized simplicial complexes (spaces obtained by stitching together points, lines, triangles, tetrahedrons, and so forth), which are still singled out in the second edition as a major example of CW complexes. The author uses CW complexes to classify one-dimensional manifolds, and, in the next chapter, classifies compact, connected surfaces (two-dimensional manifolds). The latter discussion is comparable to the first chapter of Massey’s book Algebraic Topology: An Introduction.
The third part of the book (chapters 7 through 12) constitutes an introduction to the (first) fundamental group (and related topics) that is also roughly comparable in content to the treatment in Massey, as well as to the second half of Munkres’ well-known introductory topology text, though pitched at what seemed to me a bit higher level and with more of an emphasis on manifolds. Topics discussed in these chapters include the definition of the fundamental group, the fundamental group of the circle and degree theory, deformation retracts, the Siefert-Van Kampen theorem, and covering spaces. There is also a chapter on topics in group theory (free groups, etc.) that are useful in algebraic topology and which may not have been part of a student’s prior algebra background. Higher homotopy groups are briefly mentioned in a one-page discussion, but their study is not pursued.
The final part of the book, consisting of one chapter, constitutes an introduction to homology groups, a topic not discussed in either Massey or Munkres. After a nice introduction explaining in intuitive terms what homology does that homotopy does not, the author proceeds to a careful definition of the homology groups, establishes the relationship between the first homology group and the fundamental group, discusses the Mayer-Vietoris sequence, calculates the homology groups of a number of examples, and provides some applications of homology. A final five-page discussion provides a quick introduction to cohomology.
The author has, I think, more than satisfactorily fulfilled his objective of integrating a study of manifolds into an introductory course in general and algebraic topology. This text is well-organized and clearly written, with a good blend of motivational discussion and mathematical rigor. There are numerous exercises of varying difficulty, including “exercises” that appear throughout the body of the text and which should be looked at as they are encountered, and the more substantial “problems” that appear at the end of each chapter; the latter include a number of well-known results such as the Fundamental Theorem of Algebra, the Borsuk-Ulam theorem, invariance of dimension, the Brouwer Fixed Point theorem, and the fact that the topologist’s sine curve is connected but not path connected. Any student who has gone through this book should be well-prepared to pursue the study of differential geometry (for which the author’s two “sequels”, one on differentiable manifolds and one on Riemannian geometry, would be good choices) or go on to more advanced topics in algebraic topology such as the higher homotopy groups or cohomology, as in, say, Hatcher’s book.
Mark Hunacek is a lecturer at Iowa State University. After near-simultaneous acquisitions of both a PhD and a wife, he solved the “two body problem” in his family by going to law school and then becoming an Assistant Attorney General for the state of Iowa while his wife pursued a career as a mathematics professor. He is happy to report, however, that he has now retired from the practice of law and returned to the fold of mathematics teaching (but he teaches a course in engineering law for old time’s sake.)