The geometric topology of 3-manifolds. (English) Zbl 0535.57001

The author’s mathematical personality permeates this book. If a reader is charmed by a kind of down home, friendly style then that reader will enjoy learning from, or reviewing material in, this book. If a reader needs formality and is distressed by an author taking a casual approach to serious matters, then that reader may be put off by the book. Examples of the sort of sentences that epitomize the style are, p. 203 ”...if we use the term handle without qualification, we mean it to be like the handle of a bucket” - or p. 150 ”The reason is that if someone cuts some small mutually disjoint holes in a 2-sphere and someone else does the same, then what is left might not be a 2-sphere with small mutually disjoint holes but rather a 2-sphere with big holes.” or p. 174 ”In transferring from shelling to collapsing one may just be getting out of one sinkhold (sic) and into another.”
All this is not to say the proofs in the book are loose, they definitely are not. Where the author says he will prove something, he does, generally directly and with great emphasis on giving the reader understanding. Where details are omitted it is done gracefully and with warning. Motivation is always present in proofs, and the conversational style makes that natural.
This is an outstanding book for graduate students and a good one for workers in the field, it is well referenced and the author has a sense of history and a desire to place results in a context.
The book contains many of the important theorems about Euclidean 3-space from the point of view of combinatorial topology and that space receives most attention. Material about other 3-manifolds is concentrated in Dehn’s Lemma (Chapter 16), the Loop Theorem (Chapter 17), and triangulation and approximation of continuous maps by PL maps (Chapter 18). There are nice elementary treatments of covering spaces (Chapter 15) and the fundamental group (Chapter 6).
Results from Chapter 11 on intersections of spheres and 1-simplexes and Chapter 12 on intersections of surfaces with skeletons lead to one of the book’s high points, Chapter 13. This is an especially well done piece with a gentle (but rigorous) proof of the side approximation theorem.
There are three chapters on the Schoenflies Theorem (Chapters 3, 5, 14) and one on wild spheres (Chapter 4).
The middle Chapters (7, 8, 9, 10) deal respectively with mappings onto spheres, linking, separation and pulling back feelers, while Chapters 1 and 2 introduce the reader to planar complexes and their mappings.
Reviewer: L.Neuwirth


57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57Q15 Triangulating manifolds
57M10 Covering spaces and low-dimensional topology
57Q55 Approximations in PL-topology