How Surfaces Intersect In Space: An Introduction To Topology (2nd Edition)
Seiten
1995
|
2nd Revised edition
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-2066-2 (ISBN)
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-2066-2 (ISBN)
These pictures illustrate standard examples in low dimensional topology. The text starts at the most basic level (the intersection of coordinate planes) and gives hands-on constructions of some beautiful examples in topology, including Poincare's example of a homology sphere, and knotted surfaces.
This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.
This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.
Classification of orientable surfaces, and the meaning of space; examples of non-orientable surfaces including models of the projective plane and the Klein bottle; how curves fit on surfaces and gives a general discussion of knotted strings in space; some examples of other 3-dimensional spaces - the 3-dimensional sphere, lens spaces, the quaternionic projective space; movie techniques of studying surfaces in 4-dimensions - how to move among the standard examples of Klein bottles, "movie move" decomposition of turning the 2-sphere inside out; higher dimensional spaces.
| Erscheint lt. Verlag | 1.5.1995 |
|---|---|
| Reihe/Serie | Series on Knots & Everything ; 2 |
| Verlagsort | Singapore |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 981-02-2066-9 / 9810220669 |
| ISBN-13 | 978-981-02-2066-2 / 9789810220662 |
| Zustand | Neuware |
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