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Non-metrisable manifolds. (English) Zbl 1336.57031

Singapore: Springer (ISBN 978-981-287-256-2/hbk; 978-981-287-257-9/ebook). xvi, 203 p. (2014).
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. This concept, originated in B. Riemann’s doctoral thesis in the middle of the nineteenth century, is central to many parts of geometry and modern mathematical physics, because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Formally, a manifold is a connected, Hausdorff, topological space \(M\) such that there is some fixed positive integer \(n\) (called the dimension of \(M\)) such that for each point \(x\in M\) there is an open neighbourhood \(U \subseteq M\) of \(x\) and an embedding which maps \(U\) to an open subset of \(\mathbb R^n\). However, this is a rather general definition of a manifold. In most of textbooks on topology and geometry, manifolds have additional structure. For example, to exclude manifolds which are in some sense “too large”, second countability is a “common” requirement, and this condition forces a manifold to be metrizable. In contrast to metrizable manifolds which have captured a lot of attention, the study of non-metrizable manifolds had not got much attention until the work of M. E. Rudin [Houston J. Math. 5, 249–252 (1979; Zbl 0418.03036)], as well as M. E. Rudin and P. Zenor [ibid. 2, 129–134 (1976; Zbl 0315.54028)]. As mentioned by the author in the preface, what delayed the study of non-metrizable manifolds is the need for some main tools, one of which is Set Theory.
The author of this book is one of the leading experts on non-metrizable manifolds, and has focused his research on the subject in the past two decades. This book collects the research work of the author, his graduate students and collaborators on non-metrizable manifolds in nine chapters. Chapter 1 introduces manifolds and provides examples of non-metrizable manifolds; Chapter 2 explores the frontier between metrizable and non-metrizable manifolds, and 119 topological conditions each of which is equivalent to metrizability of manifolds are listed; Chapter 3 contains some useful geometric tools like Morton Brown’s result on open \(n\)-cells and his collaring theorem; Chapter 4 deals with a type of manifolds introduced by Peter Nyikos and provides a proof of his Bagpipe theorem; Chapter 5 studies dynamics on non-metrizable manifolds; Chapter 6 addresses the old question whether perfectly normal manifolds are metrizable, and contains the construction of the Rudin-Zenor surface under CH, which is perfectly normal but not metrizable, and also contains Mary E. Rudin’s proof that perfectly normal manifolds are metrizable under MA+ \(\neg\)CH; Chapter 7 looks at differential structures, especially on the long line and the long plane; Chapter 8 presents some results on foliations in the non-metrizable context; Chapter 9 explores what can happen if we relax the hypothesis that manifolds must be Hausdorff.
The book is well-organized with two appendices: the first one collects all necessary topological results, and the second one provides preliminaries on set theory. In addition, each chapter has its own abstract and references. To the best knowledge of the reviewer, this book is so-far the only monograph in the literature which gives a comprehensive treatment on non-metrizable manifolds. It is recommended to those readers who have general knowledge on manifolds as topological objects and are curious about what happens beyond the wall of metrizability.

MSC:

57N99 Topological manifolds
54E35 Metric spaces, metrizability
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