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E-raamat: Non-metrisable Manifolds

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  • Ilmumisaeg: 04-Dec-2014
  • Kirjastus: Springer Verlag, Singapore
  • Keel: eng
  • ISBN-13: 9789812872579
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Non-metrisable ManifoldsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 04-Dec-2014
  • Kirjastus: Springer Verlag, Singapore
  • Keel: eng
  • ISBN-13: 9789812872579
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Manifolds fall naturally into two classes depending on whether they can be fitted with a distance measuring function or not. The former, metrisable manifolds, and especially compact manifolds, have been intensively studied by topologists for over a century, whereas the latter, non-metrisable manifolds, are much more abundant but have a more modest history, having become of increasing interest only over the past 40 years or so. The first book on this topic, this book ranges from criteria for metrisability, dynamics on non-metrisable manifolds, Nyikos’s Bagpipe Theorem and whether perfectly normal manifolds are metrisable to structures on manifolds, especially the abundance of exotic differential structures and the dearth of foliations on the long plane. A rigid foliation of the Euclidean plane is described. This book is intended for graduate students and mathematicians who are curious about manifolds beyond the metrisability wall, and especially the use of Set Theory as a tool.

Arvustused

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. 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. (Jiling Cao, zbMATH 1336.57031, 2016)

First of its kind, this volume by Gauld (Univ. of Auckland, New Zealand) both synthesizes and improves upon the journal literature, demonstrating clearly that non-metrizable manifolds admit profitable study from a variety of vantages (e.g., set theory, differential topology) and exhibitrich and surprising behaviors, using theory built on, but hardly reducible to, the metrizable case. Summing Up: Highly recommended. Upper-division undergraduates through professionals/practitioners. (D. V. Feldman, Choice, Vol. 53 (2), October, 2015)

1 Topological Manifolds
1(20)
1.1 Manifolds and Some Examples
1(5)
1.2 Basic Properties of the Long Line
6(6)
1.3 Some Standard Manifold Constructions
12(9)
References
19(2)
2 Edge of the World: When Are Manifolds Metrisable?
21(16)
2.1 Definitions
21(6)
2.2 Conditions Equivalent to Metrisability
27(10)
References
34(3)
3 Geometric Tools
37(12)
3.1 A Countable Union of Open n-Cells
37(5)
3.2 Collaring the Boundary of a Metrisable Manifold
42(4)
3.3 Handlebodies
46(3)
References
48(1)
4 Type I Manifolds and the Bagpipe Theorem
49(14)
4.1 Type I Manifolds
49(3)
4.2 The Nyikos Bagpipe Theorem
52(7)
4.3 The Cardinality of ω-Bounded, Non-metrisable Manifolds
59(4)
References
62(1)
5 Homeomorphisms and Dynamics on Non-metrisable Manifolds
63(24)
5.1 Continuous Flows
63(2)
5.2 Discrete Flows
65(22)
References
86(1)
6 Are Perfectly Normal Manifolds Metrisable?
87(14)
6.1 A Perfectly Normal Manifold Which Is not Metrisable
87(8)
6.2 Every Perfectly Normal Manifold Is Metrisable
95(6)
References
100(1)
7 Smooth Manifolds
101(28)
7.1 Smooth Manifolds
101(4)
7.2 Smooth Structures on the Long Line
105(10)
7.3 Exotic Differential Structures on the Long Plane
115(14)
References
127(2)
8 Foliations on Non-metrisable Manifolds
129(24)
8.1 Examples of Foliations
130(6)
8.2 Black Holes
136(6)
8.3 Foliating a Long Cylinder
142(3)
8.4 Foliating the Long Plane
145(8)
References
152(1)
9 Non-Hausdorff Manifolds and Foliations
153(14)
9.1 Non-Hausdorff Manifolds
153(8)
9.2 Foliations of the Plane
161(6)
References
165(2)
Appendix A Topological Preliminaries 167(18)
Appendix B Set Theory Preliminaries 185(14)
Index 199
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