A Topological Aperitif

163 Pages·2009·0.732 MB·English

by  David Jordan, Stephen Huggett (auth.)

A Topological Aperitif
Revised Edition

Stephen Huggett David Jordan
·
A Topological Aperitif
Revised Edition
123
StephenHuggett DavidJordan
Plymouth,UK Hull,UK
ToAnneandDinh
ISBN978-1-84800-912-7 e-ISBN978-1-84800-913-4
DOI10.1007/978-1-84800-913-4
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Foreword
Topologyhasbeenreferredtoas“rubber-sheetgeometry”.Thenameisapt,for
the subject is concerned with properties of an object that would be preserved,
no matter how much it is stretched, squashed, or distorted, so long as it is not
inanywaytornapartorgluedtogether.One’sfirstreactionmightbethatsuch
animprecise-soundingsubjectcouldhardlybepartofseriousmathematics,and
wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlour
games. This reaction could hardly be further from the truth. Topology is one
of the most important and broad-ranging disciplines of modern mathematics.
It is a subject of great precision and of breadth of development. It has vastly
many applications, some of great importance, ranging from particle physics to
cosmology, and from hydrodynamics to algebra and number theory.
It is also a subject of great beauty and depth. To appreciate something of
this,itisnotnecessarytodelveintothemoreobscureaspectsofmathematical
formalism. For topology is, at least initially, a very visual subject. Some of its
concepts apply to spaces of large numbers of dimensions, and therefore do not
easily submit to reasoning that depends upon direct pictorial representation.
But even in such cases, important insights can be obtained from the visual pe-
rusal of a simple geometrical configuration. Although much modern topology
dependsuponfinelytunedabstractalgebraicmachineryofgreatmathematical
sophistication, the underlying ideas are often very simple and can be appreci-
ated by the examination of properties of elementary-looking drawings.
Wefindmanyexamplesofthiskindofthinginthisbook.Thereareagreat
many diagrams, carefully chosen so as to bring out, in a directly visual way,
most of the basic ingredients of topological theory. It provides a marvellous
introduction to the subject, with many different tastes of ideas that can be
appreciated by a reader without much in the way of mathematical sophistica-
tion. The reader who desires to follow up these fascinating ideas will stand in
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vi Foreword
an excellent position to pursue the subject further, having mastered the basic
techniques that are introduced here.
The authors of this work, Stephen Huggett and David Jordan, both have
excellent credentials for explaining the beauties of this curiously austere but
potentially enormously general form of geometry. Some 20 years ago, Stephen
Huggettwasagraduatestudentofmine,andhealwayshadaparticularflairfor
conveyingtheexcitementthathehimselffeltforthemagnificenceofgeometrical
andtopologicalformsofargument.DavidJordanisknowntomeasthecreator
of some beautifully constructed and ingeniously precise geometrical shapes.
Both authors are clearly well placed to do the job that they have set out here
to do, and in this I believe that they have succeeded excellently.
Roger Penrose
Mathematical Institute, Oxford
Preface
Logic merely sanctions
the conquests of the intuition.
Jacques Hadamard
Topology is geometry without distance or angle. The geometrical objects of
study, not rigid but rather made of rubber or elastic, are especially stretchy.
We want to present mathematics that is mind-stretching and magic, of a
style that is conceptual and geometric rather than formulaic. In doing so we
hopetowhetthereader’sappetiteforthiswayofthinking,whichisatthesame
time very old and very modern. It started with classical Greek geometry and
isstillakeypartofcurrentmathematicalresearch,whichisespeciallylivelyin
geometry and topology. Indeed, just as in classical Greece, our understanding
of the physical universe depends upon this geometrical thinking.
The heart of the book is in the first five chapters: homeomorphisms, sur-
faces, and polyhedra. Although these ideas are broadly pitched at the level of
a second year undergraduate, the authors expect a tenacious mind with much
less background to grasp them. The arguments of Chapter 6, still geometri-
cal in style, add strength to the earlier chapters. This is not a book of pure
geometry, as is Euclid, but rests upon the fine structure of the real number
system.Theseunderpinningsaremostlyextractedfromtheearlychaptersand
collectedinAppendixA.AppendixBgivesafleetingglimpseintoknottheory,
introducing the Jones polynomial. Further breadth is given in Appendix C, in
which we sketch the curious and instructive early history of topology.
The main ideas are illuminated by a wide variety of geometrical examples
that we hope will fascinate and intrigue. Although elementary, the mathemat-
ics in this book is sharp and subtle, and will not be properly grasped without
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viii Preface
serious attempts at the exercises, the essential challenge of which may be un-
done by a premature glimpse of an illustrated solution. If you want to be a
pianist you don’t just read music and listen to it, you play it.
Severalpeoplehavebeenextremelyhelpfultousinwritingthisbook.Weare
very grateful to Colin Christopher, Neil Gordon, Charlotte Malcolmson, Dinh
Phung, and Hannah Walker for all their work. Also, we are deeply indebted to
John Moran for his patience with and dedication to the pictures.
For this revised edition we have made a number of corrections to the text
and the figures, throughout the book, and we have written a short but com-
pletelynewsectionattheendofchapterfour.TheKleinbottlecanbethought
of as a sphere with a “Klein handle”. We illustrate how, given a sphere with
anynumberofordinaryhandlesandatleastoneKleinhandle,alltheordinary
handles can be converted into Klein handles. This is a part of the important
“Classification Theorem” for surfaces.
Contents
Foreword ....................................................... v
Preface ......................................................... vii
1. Homeomorphic Sets ........................................ 1
2. Topological Properties...................................... 15
3. Equivalent Subsets ......................................... 25
4. Surfaces and Spaces ........................................ 51
5. Polyhedra .................................................. 69
6. Winding Number........................................... 93
A. Continuity..................................................105
B. Knots ......................................................113
C. History .....................................................121
D. Solutions ...................................................127
Bibliography....................................................149
Index...........................................................151
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