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**Decompositions of manifolds.**
*(English)*
Zbl 0608.57002

Pure and Applied Mathematics, 124. Orlando etc.: Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. XI, 317 p.; $ 55.00; £46.00 (1986).

This is a textbook for a graduate course on (upper semicontinuous) decompositions (or partitions) of manifolds into cell-like compacta, or equivalently, on cell-like mappings on manifolds. Decomposition theory is a vital part of modern geometric topology and its history begins in the early 1920’s (it is associated with the names of R. L. Moore and G. T. Whyburn). So far there has been no systematic treatment of this important subject although in the last decades its methods helped in resolving some of the most challenging problems of geometric topology.

Let us briefly mention some of them: (1) In 1957 R. H. Bing constructed his celebrated dogbone space - a decomposition of \({\mathbb{R}}^ 3\) into points and tame arcs such that the decomposition space is topologically different from \({\mathbb{R}}^ 3\). He then proved that this space is a factor of \({\mathbb{R}}^ 4\) and thus provided the first example of a nonmanifold factor of a Euclidean space. (2) In 1977 R. D. Edwards made a dramatic breakthrough when he obtained a very powerful result which characterizes the cell-like mappings on higher-dimensional manifolds (\(\geq 5)\) that can be approximated by homeomorphisms. As its most important application one gets a beautiful simple characterization of higher-dimensional manifolds in terms of their elementary topological properties by invoking F. S. Quinn’s work on resolutions. (3) Finally, in 1981 M. H. Freedman developed a remarkably elegant classification of simply connected 4- manifolds. Again, one of the key arguments invokes decomposition theory techniques.

The author is one of the most active people working in decomposition theory and with over 80 publications he has a very distinguished record in this and related areas. He has an excellent teaching reputation and in his many lectures he has demonstrated a wide knowledge of the various techniques in geometric topology, many of which he has invented. He is unusually adept at distinguishing the main theme from the many other thoughts and ideas and in communicating this to his audience. He started to work on this book about a decade ago and he has spent several years organizing and systematizing the subject of cell-like decompositions. Its preliminary version was circulated among his colleagues and many of us used it in class - with great success.

The book starts with basic terminology and explores the elementary properties of decompositions. After a quick overview of monotone decompositions it enters into a study of various notions of shrinkability and eventually proves them to be equivalent. It then presents Bing’s shrinking criterion, by now a classical device for recognizing the (non)singular nature of cell-like quotients of manifolds. Its power is illustrated with several examples - this is a nice account of Bing topology.

After a technical chapter, devoted to the study of properties by the typical decompositions, comes the proof of the fundamental result of manifold decomposition theory, Edwards’ cell-like approximation theorem and its many applications, mostly to the products of decomposition spaces with the real line or with another such space. Included is also a description of spinning, a very useful method for generating decompositions of higher-dimensional manifolds. In the sequel, many pathological, i.e. nonshrinkable cellular decompositions are constructed (such exist in dimensions \(\geq 4\) only). The book concludes with applications of decomposition theory to the rest of geometric topology. Throughout the text there are exercises, problems and questions, ranging from simple to quite challenging.

The author’s book is excellent and it was long overdue. It is a most welcome contribution to the mathematical literature and I strongly recommend it to anyone who wishes to learn about decompositions of manifolds.

Let us briefly mention some of them: (1) In 1957 R. H. Bing constructed his celebrated dogbone space - a decomposition of \({\mathbb{R}}^ 3\) into points and tame arcs such that the decomposition space is topologically different from \({\mathbb{R}}^ 3\). He then proved that this space is a factor of \({\mathbb{R}}^ 4\) and thus provided the first example of a nonmanifold factor of a Euclidean space. (2) In 1977 R. D. Edwards made a dramatic breakthrough when he obtained a very powerful result which characterizes the cell-like mappings on higher-dimensional manifolds (\(\geq 5)\) that can be approximated by homeomorphisms. As its most important application one gets a beautiful simple characterization of higher-dimensional manifolds in terms of their elementary topological properties by invoking F. S. Quinn’s work on resolutions. (3) Finally, in 1981 M. H. Freedman developed a remarkably elegant classification of simply connected 4- manifolds. Again, one of the key arguments invokes decomposition theory techniques.

The author is one of the most active people working in decomposition theory and with over 80 publications he has a very distinguished record in this and related areas. He has an excellent teaching reputation and in his many lectures he has demonstrated a wide knowledge of the various techniques in geometric topology, many of which he has invented. He is unusually adept at distinguishing the main theme from the many other thoughts and ideas and in communicating this to his audience. He started to work on this book about a decade ago and he has spent several years organizing and systematizing the subject of cell-like decompositions. Its preliminary version was circulated among his colleagues and many of us used it in class - with great success.

The book starts with basic terminology and explores the elementary properties of decompositions. After a quick overview of monotone decompositions it enters into a study of various notions of shrinkability and eventually proves them to be equivalent. It then presents Bing’s shrinking criterion, by now a classical device for recognizing the (non)singular nature of cell-like quotients of manifolds. Its power is illustrated with several examples - this is a nice account of Bing topology.

After a technical chapter, devoted to the study of properties by the typical decompositions, comes the proof of the fundamental result of manifold decomposition theory, Edwards’ cell-like approximation theorem and its many applications, mostly to the products of decomposition spaces with the real line or with another such space. Included is also a description of spinning, a very useful method for generating decompositions of higher-dimensional manifolds. In the sequel, many pathological, i.e. nonshrinkable cellular decompositions are constructed (such exist in dimensions \(\geq 4\) only). The book concludes with applications of decomposition theory to the rest of geometric topology. Throughout the text there are exercises, problems and questions, ranging from simple to quite challenging.

The author’s book is excellent and it was long overdue. It is a most welcome contribution to the mathematical literature and I strongly recommend it to anyone who wishes to learn about decompositions of manifolds.

Reviewer: D.Repovš

### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |

57N60 | Cellularity in topological manifolds |

54B15 | Quotient spaces, decompositions in general topology |