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Topology: A Very Short Introduction
Topology: A Very Short Introduction
Richard Earl
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How is a subway map different from other maps? What makes a knot knotted? What makes the Möbius strip onesided? These are questions of topology, the mathematical study of properties preserved by twisting or stretching objects. In the 20th century topology became as broad and fundamental as algebra and geometry, with important implications for science, especially physics.
In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eyeopening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field.
ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocketsized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eyeopening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field.
ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocketsized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
Категории:
Год:
2020
Издательство:
Oxford University Press, USA
Язык:
english
Страницы:
168
ISBN 10:
0198832680
ISBN 13:
9780198832683
Серии:
Very Short Introductions
Файл:
EPUB, 4,44 MB
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Ключевые фразы
function^{146}
points^{121}
functions^{107}
continuous^{105}
topology^{104}
knot^{101}
numbers^{101}
torus^{94}
plane^{91}
loop^{82}
theorem^{80}
edges^{80}
connected^{78}
surface^{77}
knots^{73}
circle^{72}
inputs^{68}
metric^{66}
example^{65}
sets^{65}
surfaces^{63}
vertices^{58}
euler number^{57}
loops^{52}
topological^{50}
spaces^{49}
square^{49}
input^{48}
polynomial^{45}
mathematics^{45}
outputs^{45}
curve^{44}
sphere^{44}
interval^{43}
glued^{43}
faces^{41}
david^{41}
fundamental^{40}
formula^{38}
continuity^{38}
chapter^{37}
trefoil^{37}
ball^{36}
value^{35}
sequence^{35}
graph^{34}
definition^{33}
output^{33}
complex^{33}
crossings^{33}
metrics^{32}
dimensional^{30}
crossing^{30}
unknot^{29}
around the circle^{29}
equivalent^{29}
continuous functions^{28}
axis^{28}
alexander^{28}
examples^{27}
Связанные Буклисты
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Topology: A Very Short Introduction VERY SHORT INTRODUCTIONS are for anyone wanting a stimulating and accessible way into a new subject. They are written by experts, and have been translated into more than 45 different languages. The series began in 1995, and now covers a wide variety of topics in every discipline. The VSI library currently contains over 600 volumes—a Very Short Introduction to everything from Psychology and Philosophy of Science to American History and Relativity—and continues to grow in every subject area. Very Short Introductions available now: ABOLITIONISM Richard S. Newman THE ABRAHAMIC RELIGIONS Charles L. Cohen ACCOUNTING Christopher Nobes ADAM SMITH Christopher J. Berry ADOLESCENCE Peter K. Smith ADVERTISING Winston Fletcher AESTHETICS Bence Nanay AFRICAN AMERICAN RELIGION Eddie S. Glaude Jr AFRICAN HISTORY John Parker and Richard Rathbone AFRICAN POLITICS Ian Taylor AFRICAN RELIGIONS Jacob K. Olupona AGEING Nancy A. Pachana AGNOSTICISM Robin Le Poidevin AGRICULTURE Paul Brassley and Richard Soffe ALEXANDER THE GREAT Hugh Bowden ALGEBRA Peter M. Higgins AMERICAN CULTURAL HISTORY Eric Avila AMERICAN FOREIGN RELATIONS Andrew Preston AMERICAN HISTORY Paul S. Boyer AMERICAN IMMIGRATION David A. Gerber AMERICAN LEGAL HISTORY G. Edward White AMERICAN NAVAL HISTORY Craig L. Symonds AMERICAN POLITICAL HISTORY Donald Critchlow AMERICAN POLITICAL PARTIES AND ELECTIONS L. Sandy Maisel AMERICAN POLITICS Richard M. Valelly THE AMERICAN PRESIDENCY Charles O. Jones THE AMERICAN REVOLUTION Robert J. Allison AMERICAN SLAVERY Heather Andrea Williams THE AMERICAN WEST Stephen Aron AMERICAN WOMEN’S HISTORY Susan Ware ANAESTHESIA Aidan O’Donnell ANALYTIC PHILOSOPHY Michael Beaney ANARCHISM Colin Ward ANCIENT ASSYRIA Karen Radner ANCIENT EGYPT Ian Shaw ANCIENT EGYPTIAN ART AND ARCHITECTURE Christina Riggs ANCIENT GREECE Paul Cartledge THE ANCIENT NEAR EAST Amanda H. Podany ANCIENT PHILOSOPHY Julia Annas ANCIENT WARFARE Harry Sidebottom ANGELS David Alb; ert Jones ANGLICANISM Mark Chapman THE ANGLOSAXON AGE John Blair ANIMAL BEHAVIOUR Tristram D. Wyatt THE ANIMAL KINGDOM Peter Holland ANIMAL RIGHTS David DeGrazia THE ANTARCTIC Klaus Dodds ANTHROPOCENE Erle C. Ellis ANTISEMITISM Steven Beller ANXIETY Daniel Freeman and Jason Freeman THE APOCRYPHAL GOSPELS Paul Foster APPLIED MATHEMATICS Alain Goriely ARCHAEOLOGY Paul Bahn ARCHITECTURE Andrew Ballantyne ARISTOCRACY William Doyle ARISTOTLE Jonathan Barnes ART HISTORY Dana Arnold ART THEORY Cynthia Freeland ARTIFICIAL INTELLIGENCE Margaret A. Boden ASIAN AMERICAN HISTORY Madeline Y. Hsu ASTROBIOLOGY David C. Catling ASTROPHYSICS James Binney ATHEISM Julian Baggini THE ATMOSPHERE Paul I. Palmer AUGUSTINE Henry Chadwick AUSTRALIA Kenneth Morgan AUTISM Uta Frith AUTOBIOGRAPHY Laura Marcus THE AVANT GARDE David Cottington THE AZTECS Davíd Carrasco BABYLONIA Trevor Bryce BACTERIA Sebastian G. B. Amyes BANKING John Goddard and John O. S. Wilson BARTHES Jonathan Culler THE BEATS David Sterritt BEAUTY Roger Scruton BEHAVIOURAL ECONOMICS Michelle Baddeley BESTSELLERS John Sutherland THE BIBLE John Riches BIBLICAL ARCHAEOLOGY Eric H. Cline BIG DATA Dawn E. Holmes BIOGRAPHY Hermione Lee BIOMETRICS Michael Fairhurst BLACK HOLES Katherine Blundell BLOOD Chris Cooper THE BLUES Elijah Wald THE BODY Chris Shilling THE BOOK OF COMMON PRAYER Brian Cummings THE BOOK OF MORMON Terryl Givens BORDERS Alexander C. Diener and Joshua Hagen THE BRAIN Michael O’Shea BRANDING Robert Jones THE BRICS Andrew F. Cooper THE BRITISH CONSTITUTION Martin Loughlin THE BRITISH EMPIRE Ashley Jackson BRITISH POLITICS Anthony Wright BUDDHA Michael Carrithers BUDDHISM Damien Keown BUDDHIST ETHICS Damien Keown BYZANTIUM Peter Sarris C. S. LEWIS James Como CALVINISM Jon Balserak CANCER Nicholas James CAPITALISM James Fulcher CATHOLICISM Gerald O’Collins CAUSATION Stephen Mumford and Rani Lill Anjum THE CELL Terence Allen and Graham Cowling THE CELTS Barry Cunliffe CHAOS Leonard Smith CHARLES DICKENS Jenny Hartley CHEMISTRY Peter Atkins CHILD PSYCHOLOGY Usha Goswami CHILDREN’S LITERATURE Kimberley Reynolds CHINESE LITERATURE Sabina Knight CHOICE THEORY Michael Allingham CHRISTIAN ART Beth Williamson CHRISTIAN ETHICS D. Stephen Long CHRISTIANITY Linda Woodhead CIRCADIAN RHYTHMS Russell Foster and Leon Kreitzman CITIZENSHIP Richard Bellamy CIVIL ENGINEERING David Muir Wood CLASSICAL LITERATURE William Allan CLASSICAL MYTHOLOGY Helen Morales CLASSICS Mary Beard and John Henderson CLAUSEWITZ Michael Howard CLIMATE Mark Maslin CLIMATE CHANGE Mark Maslin CLINICAL PSYCHOLOGY Susan Llewelyn and Katie Aafjesvan Doorn COGNITIVE NEUROSCIENCE Richard Passingham THE COLD WAR Robert McMahon COLONIAL AMERICA Alan Taylor COLONIAL LATIN AMERICAN LITERATURE Rolena Adorno COMBINATORICS Robin Wilson COMEDY Matthew Bevis COMMUNISM Leslie Holmes COMPARATIVE LITERATURE Ben Hutchinson COMPLEXITY John H. Holland THE COMPUTER Darrel Ince COMPUTER SCIENCE Subrata Dasgupta CONCENTRATION CAMPS Dan Stone CONFUCIANISM Daniel K. Gardner THE CONQUISTADORS Matthew Restall and Felipe FernándezArmesto CONSCIENCE Paul Strohm CONSCIOUSNESS Susan Blackmore CONTEMPORARY ART Julian Stallabrass CONTEMPORARY FICTION Robert Eaglestone CONTINENTAL PHILOSOPHY Simon Critchley COPERNICUS Owen Gingerich CORAL REEFS Charles Sheppard CORPORATE SOCIAL RESPONSIBILITY Jeremy Moon CORRUPTION Leslie Holmes COSMOLOGY Peter Coles COUNTRY MUSIC Richard Carlin CRIME FICTION Richard Bradford CRIMINAL JUSTICE Julian V. Roberts CRIMINOLOGY Tim Newburn CRITICAL THEORY Stephen Eric Bronner THE CRUSADES Christopher Tyerman CRYPTOGRAPHY Fred Piper and Sean Murphy CRYSTALLOGRAPHY A. M. Glazer THE CULTURAL REVOLUTION Richard Curt Kraus DADA AND SURREALISM David Hopkins DANTE Peter Hainsworth and David Robey DARWIN Jonathan Howard THE DEAD SEA SCROLLS Timothy H. Lim DECADENCE David Weir DECOLONIZATION Dane Kennedy DEMOCRACY Bernard Crick DEMOGRAPHY Sarah Harper DEPRESSION Jan Scott and Mary Jane Tacchi DERRIDA Simon Glendinning DESCARTES Tom Sorell DESERTS Nick Middleton DESIGN John Heskett DEVELOPMENT Ian Goldin DEVELOPMENTAL BIOLOGY Lewis Wolpert THE DEVIL Darren Oldridge DIASPORA Kevin Kenny DICTIONARIES Lynda Mugglestone DINOSAURS David Norman DIPLOMACY Joseph M. Siracusa DOCUMENTARY FILM Patricia Aufderheide DREAMING J. Allan Hobson DRUGS Les Iversen DRUIDS Barry Cunliffe DYNASTY Jeroen Duindam DYSLEXIA Margaret J. Snowling EARLY MUSIC Thomas Forrest Kelly THE EARTH Martin Redfern EARTH SYSTEM SCIENCE Tim Lenton ECONOMICS Partha Dasgupta EDUCATION Gary Thomas EGYPTIAN MYTH Geraldine Pinch EIGHTEENTH‑CENTURY BRITAIN Paul Langford THE ELEMENTS Philip Ball EMOTION Dylan Evans EMPIRE Stephen Howe ENERGY SYSTEMS Nick Jenkins ENGELS Terrell Carver ENGINEERING David Blockley THE ENGLISH LANGUAGE Simon Horobin ENGLISH LITERATURE Jonathan Bate THE ENLIGHTENMENT John Robertson ENTREPRENEURSHIP Paul Westhead and Mike Wright ENVIRONMENTAL ECONOMICS Stephen Smith ENVIRONMENTAL ETHICS Robin Attfield ENVIRONMENTAL LAW Elizabeth Fisher ENVIRONMENTAL POLITICS Andrew Dobson EPICUREANISM Catherine Wilson EPIDEMIOLOGY Rodolfo Saracci ETHICS Simon Blackburn ETHNOMUSICOLOGY Timothy Rice THE ETRUSCANS Christopher Smith EUGENICS Philippa Levine THE EUROPEAN UNION Simon Usherwood and John Pinder EUROPEAN UNION LAW Anthony Arnull EVOLUTION Brian and Deborah Charlesworth EXISTENTIALISM Thomas Flynn EXPLORATION Stewart A. Weaver EXTINCTION Paul B. Wignall THE EYE Michael Land FAIRY TALE Marina Warner FAMILY LAW Jonathan Herring FASCISM Kevin Passmore FASHION Rebecca Arnold FEDERALISM Mark J. Rozell and Clyde Wilcox FEMINISM Margaret Walters FILM Michael Wood FILM MUSIC Kathryn Kalinak FILM NOIR James Naremore THE FIRST WORLD WAR Michael Howard FOLK MUSIC Mark Slobin FOOD John Krebs FORENSIC PSYCHOLOGY David Canter FORENSIC SCIENCE Jim Fraser FORESTS Jaboury Ghazoul FOSSILS Keith Thomson FOUCAULT Gary Gutting THE FOUNDING FATHERS R. B. Bernstein FRACTALS Kenneth Falconer FREE SPEECH Nigel Warburton FREE WILL Thomas Pink FREEMASONRY Andreas Önnerfors FRENCH LITERATURE John D. Lyons THE FRENCH REVOLUTION William Doyle FREUD Anthony Storr FUNDAMENTALISM Malise Ruthven FUNGI Nicholas P. Money THE FUTURE Jennifer M. Gidley GALAXIES John Gribbin GALILEO Stillman Drake GAME THEORY Ken Binmore GANDHI Bhikhu Parekh GARDEN HISTORY Gordon Campbell GENES Jonathan Slack GENIUS Andrew Robinson GENOMICS John Archibald GEOFFREY CHAUCER David Wallace GEOGRAPHY JOHN Matthews and David Herbert GEOLOGY Jan Zalasiewicz GEOPHYSICS William Lowrie GEOPOLITICS Klaus Dodds GERMAN LITERATURE Nicholas Boyle GERMAN PHILOSOPHY Andrew Bowie GLACIATION David J. A. Evans GLOBAL CATASTROPHES Bill McGuire GLOBAL ECONOMIC HISTORY Robert C. Allen GLOBALIZATION Manfred Steger GOD John Bowker GOETHE Ritchie Robertson THE GOTHIC Nick Groom GOVERNANCE Mark Bevir GRAVITY Timothy Clifton THE GREAT DEPRESSION AND THE NEW DEAL Eric Rauchway HABERMAS James Gordon Finlayson THE HABSBURG EMPIRE Martyn Rady HAPPINESS Daniel M. Haybron THE HARLEM RENAISSANCE Cheryl A. Wall THE HEBREW BIBLE AS LITERATURE Tod Linafelt HEGEL Peter Singer HEIDEGGER Michael Inwood THE HELLENISTIC AGE Peter Thonemann HEREDITY John Waller HERMENEUTICS Jens Zimmermann HERODOTUS Jennifer T. Roberts HIEROGLYPHS Penelope Wilson HINDUISM Kim Knott HISTORY John H. Arnold THE HISTORY OF ASTRONOMY Michael Hoskin THE HISTORY OF CHEMISTRY William H. Brock THE HISTORY OF CHILDHOOD James Marten THE HISTORY OF CINEMA Geoffrey NowellSmith THE HISTORY OF LIFE Michael Benton THE HISTORY OF MATHEMATICS Jacqueline Stedall THE HISTORY OF MEDICINE William Bynum THE HISTORY OF PHYSICS J. L. Heilbron THE HISTORY OF TIME Leofranc Holford‑Strevens HIV AND AIDS Alan Whiteside HOBBES Richard Tuck HOLLYWOOD Peter Decherney THE HOLY ROMAN EMPIRE Joachim Whaley HOME Michael Allen Fox HOMER Barbara Graziosi HORMONES Martin Luck HUMAN ANATOMY Leslie Klenerman HUMAN EVOLUTION Bernard Wood HUMAN RIGHTS Andrew Clapham HUMANISM Stephen Law HUME A. J. Ayer HUMOUR Noël Carroll THE ICE AGE Jamie Woodward IDENTITY Florian Coulmas IDEOLOGY Michael Freeden THE IMMUNE SYSTEM Paul Klenerman INDIAN CINEMA Ashish Rajadhyaksha INDIAN PHILOSOPHY Sue Hamilton THE INDUSTRIAL REVOLUTION Robert C. Allen INFECTIOUS DISEASE Marta L. Wayne and Benjamin M. Bolker INFINITY Ian Stewart INFORMATION Luciano Floridi INNOVATION Mark Dodgson and David Gann INTELLECTUAL PROPERTY Siva Vaidhyanathan INTELLIGENCE Ian J. Deary INTERNATIONAL LAW Vaughan Lowe INTERNATIONAL MIGRATION Khalid Koser INTERNATIONAL RELATIONS Paul Wilkinson INTERNATIONAL SECURITY Christopher S. Browning IRAN Ali M. Ansari ISLAM Malise Ruthven ISLAMIC HISTORY Adam Silverstein ISOTOPES Rob Ellam ITALIAN LITERATURE Peter Hainsworth and David Robey JESUS Richard Bauckham JEWISH HISTORY David N. Myers JOURNALISM Ian Hargreaves JUDAISM Norman Solomon JUNG Anthony Stevens KABBALAH Joseph Dan KAFKA Ritchie Robertson KANT Roger Scruton KEYNES Robert Skidelsky KIERKEGAARD Patrick Gardiner KNOWLEDGE Jennifer Nagel THE KORAN Michael Cook LAKES Warwick F. Vincent LANDSCAPE ARCHITECTURE Ian H. Thompson LANDSCAPES AND GEOMORPHOLOGY Andrew Goudie and Heather Viles LANGUAGES Stephen R. Anderson LATE ANTIQUITY Gillian Clark LAW Raymond Wacks THE LAWS OF THERMODYNAMICS Peter Atkins LEADERSHIP Keith Grint LEARNING Mark Haselgrove LEIBNIZ Maria Rosa Antognazza LEO TOLSTOY Liza Knapp LIBERALISM Michael Freeden LIGHT Ian Walmsley LINCOLN Allen C. Guelzo LINGUISTICS Peter Matthews LITERARY THEORY Jonathan Culler LOCKE John Dunn LOGIC Graham Priest LOVE Ronald de Sousa MACHIAVELLI Quentin Skinner MADNESS Andrew Scull MAGIC Owen Davies MAGNA CARTA Nicholas Vincent MAGNETISM Stephen Blundell MALTHUS Donald Winch MAMMALS T. S. Kemp MANAGEMENT John Hendry MAO Delia Davin MARINE BIOLOGY Philip V. Mladenov THE MARQUIS DE SADE John Phillips MARTIN LUTHER Scott H. Hendrix MARTYRDOM Jolyon Mitchell MARX Peter Singer MATERIALS Christopher Hall MATHEMATICAL FINANCE Mark H. A. Davis MATHEMATICS Timothy Gowers MATTER Geoff Cottrell THE MEANING OF LIFE Terry Eagleton MEASUREMENT David Hand MEDICAL ETHICS Michael Dunn and Tony Hope MEDICAL LAW Charles Foster MEDIEVAL BRITAIN John Gillingham and Ralph A. Griffiths MEDIEVAL LITERATURE Elaine Treharne MEDIEVAL PHILOSOPHY John Marenbon MEMORY Jonathan K. Foster METAPHYSICS Stephen Mumford METHODISM William J. Abraham THE MEXICAN REVOLUTION Alan Knight MICHAEL FARADAY Frank A. J. L. James MICROBIOLOGY Nicholas P. Money MICROECONOMICS Avinash Dixit MICROSCOPY Terence Allen THE MIDDLE AGES Miri Rubin MILITARY JUSTICE Eugene R. Fidell MILITARY STRATEGY Antulio J. Echevarria II MINERALS David Vaughan MIRACLES Yujin Nagasawa MODERN ARCHITECTURE Adam Sharr MODERN ART David Cottington MODERN CHINA Rana Mitter MODERN DRAMA Kirsten E. ShepherdBarr MODERN FRANCE Vanessa R. Schwartz MODERN INDIA Craig Jeffrey MODERN IRELAND Senia Pašeta MODERN ITALY Anna Cento Bull MODERN JAPAN Christopher GotoJones MODERN LATIN AMERICAN LITERATURE Roberto González Echevarría MODERN WAR Richard English MODERNISM Christopher Butler MOLECULAR BIOLOGY Aysha Divan and Janice A. Royds MOLECULES Philip Ball MONASTICISM Stephen J. Davis THE MONGOLS Morris Rossabi MOONS David A. Rothery MORMONISM Richard Lyman Bushman MOUNTAINS Martin F. Price MUHAMMAD Jonathan A. C. Brown MULTICULTURALISM Ali Rattansi MULTILINGUALISM John C. Maher MUSIC Nicholas Cook MYTH Robert A. Segal NAPOLEON David Bell THE NAPOLEONIC WARS Mike Rapport NATIONALISM Steven Grosby NATIVE AMERICAN LITERATURE Sean Teuton NAVIGATION Jim Bennett NAZI GERMANY Jane Caplan NELSON MANDELA Elleke Boehmer NEOLIBERALISM Manfred Steger and Ravi Roy NETWORKS Guido Caldarelli and Michele Catanzaro THE NEW TESTAMENT Luke Timothy Johnson THE NEW TESTAMENT AS LITERATURE Kyle Keefer NEWTON Robert Iliffe NIETZSCHE Michael Tanner NINETEENTH‑CENTURY BRITAIN Christopher Harvie and H. C. G. Matthew THE NORMAN CONQUEST George Garnett NORTH AMERICAN INDIANS Theda Perdue and Michael D. Green NORTHERN IRELAND Marc Mulholland NOTHING Frank Close NUCLEAR PHYSICS Frank Close NUCLEAR POWER Maxwell Irvine NUCLEAR WEAPONS Joseph M. Siracusa NUMBERS Peter M. Higgins NUTRITION David A. Bender OBJECTIVITY Stephen Gaukroger OCEANS Dorrik Stow THE OLD TESTAMENT Michael D. Coogan THE ORCHESTRA D. Kern Holoman ORGANIC CHEMISTRY Graham Patrick ORGANIZATIONS Mary Jo Hatch ORGANIZED CRIME Georgios A. Antonopoulos and Georgios Papanicolaou ORTHODOX CHRISTIANITY A. Edward Siecienski PAGANISM Owen Davies PAIN Rob Boddice THE PALESTINIANISRAELI CONFLICT Martin Bunton PANDEMICS Christian W. McMillen PARTICLE PHYSICS Frank Close PAUL E. P. Sanders PEACE Oliver P. Richmond PENTECOSTALISM William K. Kay PERCEPTION Brian Rogers THE PERIODIC TABLE Eric R. Scerri PHILOSOPHY Edward Craig PHILOSOPHY IN THE ISLAMIC WORLD Peter Adamson PHILOSOPHY OF BIOLOGY Samir Okasha PHILOSOPHY OF LAW Raymond Wacks PHILOSOPHY OF SCIENCE Samir Okasha PHILOSOPHY OF RELIGION Tim Bayne PHOTOGRAPHY Steve Edwards PHYSICAL CHEMISTRY Peter Atkins PHYSICS Sidney Perkowitz PILGRIMAGE Ian Reader PLAGUE Paul Slack PLANETS David A. Rothery PLANTS Timothy Walker PLATE TECTONICS Peter Molnar PLATO Julia Annas POETRY Bernard O’Donoghue POLITICAL PHILOSOPHY David Miller POLITICS Kenneth Minogue POPULISM Cas Mudde and Cristóbal Rovira Kaltwasser POSTCOLONIALISM Robert Young POSTMODERNISM Christopher Butler POSTSTRUCTURALISM Catherine Belsey POVERTY Philip N. Jefferson PREHISTORY Chris Gosden PRESOCRATIC PHILOSOPHY Catherine Osborne PRIVACY Raymond Wacks PROBABILITY John Haigh PROGRESSIVISM Walter Nugent PROJECTS Andrew Davies PROTESTANTISM Mark A. Noll PSYCHIATRY Tom Burns PSYCHOANALYSIS Daniel Pick PSYCHOLOGY Gillian Butler and Freda McManus PSYCHOLOGY OF MUSIC Elizabeth Hellmuth Margulis PSYCHOPATHY Essi Viding PSYCHOTHERAPY Tom Burns and Eva BurnsLundgren PUBLIC ADMINISTRATION Stella Z. Theodoulou and Ravi K. Roy PUBLIC HEALTH Virginia Berridge PURITANISM Francis J. Bremer THE QUAKERS Pink Dandelion QUANTUM THEORY John Polkinghorne RACISM Ali Rattansi RADIOACTIVITY Claudio Tuniz RASTAFARI Ennis B. Edmonds READING Belinda Jack THE REAGAN REVOLUTION Gil Troy REALITY Jan Westerhoff THE REFORMATION Peter Marshall RELATIVITY Russell Stannard RELIGION IN AMERICA Timothy Beal THE RENAISSANCE Jerry Brotton RENAISSANCE ART Geraldine A. Johnson REPTILES T. S. Kemp REVOLUTIONS Jack A. Goldstone RHETORIC Richard Toye RISK Baruch Fischhoff and John Kadvany RITUAL Barry Stephenson RIVERS Nick Middleton ROBOTICS Alan Winfield ROCKS Jan Zalasiewicz ROMAN BRITAIN Peter Salway THE ROMAN EMPIRE Christopher Kelly THE ROMAN REPUBLIC David M. Gwynn ROMANTICISM Michael Ferber ROUSSEAU Robert Wokler RUSSELL A. C. Grayling RUSSIAN HISTORY Geoffrey Hosking RUSSIAN LITERATURE Catriona Kelly THE RUSSIAN REVOLUTION S. A. Smith THE SAINTS Simon Yarrow SAVANNAS Peter A. Furley SCEPTICISM Duncan Pritchard SCHIZOPHRENIA Chris Frith and Eve Johnstone SCHOPENHAUER Christopher Janaway SCIENCE AND RELIGION Thomas Dixon SCIENCE FICTION David Seed THE SCIENTIFIC REVOLUTION Lawrence M. Principe SCOTLAND Rab Houston SECULARISM Andrew Copson SEXUAL SELECTION Marlene Zuk and Leigh W. Simmons SEXUALITY Véronique Mottier SHAKESPEARE’S COMEDIES Bart van Es SHAKESPEARE’S SONNETS AND POEMS Jonathan F. S. Post SHAKESPEARE’S TRAGEDIES Stanley Wells SIKHISM Eleanor Nesbitt THE SILK ROAD James A. Millward SLANG Jonathon Green SLEEP Steven W. Lockley and Russell G. Foster SOCIAL AND CULTURAL ANTHROPOLOGY John Monaghan and Peter Just SOCIAL PSYCHOLOGY Richard J. Crisp SOCIAL WORK Sally Holland and Jonathan Scourfield SOCIALISM Michael Newman SOCIOLINGUISTICS John Edwards SOCIOLOGY Steve Bruce SOCRATES C. C. W. Taylor SOUND Mike Goldsmith SOUTHEAST ASIA James R. Rush THE SOVIET UNION Stephen Lovell THE SPANISH CIVIL WAR Helen Graham SPANISH LITERATURE Jo Labanyi SPINOZA Roger Scruton SPIRITUALITY Philip Sheldrake SPORT Mike Cronin STARS Andrew King Statistics David J. Hand STEM CELLS Jonathan Slack STOICISM Brad Inwood STRUCTURAL ENGINEERING David Blockley STUART BRITAIN John Morrill SUPERCONDUCTIVITY Stephen Blundell SYMMETRY Ian Stewart SYNAESTHESIA Julia Simner SYNTHETIC BIOLOGY Jamie A. Davies TAXATION Stephen Smith TEETH Peter S. Ungar TELESCOPES Geoff Cottrell TERRORISM Charles Townshend THEATRE Marvin Carlson THEOLOGY David F. Ford THINKING AND REASONING Jonathan St B. T. Evans THOMAS AQUINAS Fergus Kerr THOUGHT Tim Bayne TIBETAN BUDDHISM Matthew T. Kapstein TIDES David George Bowers and Emyr Martyn Roberts TOCQUEVILLE Harvey C. Mansfield TOPOLOGY Richard Earl TRAGEDY Adrian Poole TRANSLATION Matthew Reynolds THE TREATY OF VERSAILLES Michael S. Neiberg THE TROJAN WAR Eric H. Cline TRUST Katherine Hawley THE TUDORS John Guy TWENTIETH‑CENTURY BRITAIN Kenneth O. Morgan TYPOGRAPHY Paul Luna THE UNITED NATIONS Jussi M. Hanhimäki UNIVERSITIES AND COLLEGES David Palfreyman and Paul Temple THE U.S. CONGRESS Donald A. Ritchie THE U.S. CONSTITUTION David J. Bodenhamer THE U.S. SUPREME COURT Linda Greenhouse UTILITARIANISM Katarzyna de LazariRadek and Peter Singer UTOPIANISM Lyman Tower Sargent VETERINARY SCIENCE James Yeates THE VIKINGS Julian D. Richards VIRUSES Dorothy H. Crawford VOLTAIRE Nicholas Cronk WAR AND TECHNOLOGY Alex Roland WATER John Finney WAVES Mike Goldsmith WEATHER Storm Dunlop THE WELFARE STATE David Garland WILLIAM SHAKESPEARE Stanley Wells WITCHCRAFT Malcolm Gaskill WITTGENSTEIN A. C. Grayling WORK Stephen Fineman WORLD MUSIC Philip Bohlman THE WORLD TRADE ORGANIZATION Amrita Narlikar WORLD WAR II Gerhard L. Weinberg WRITING AND SCRIPT Andrew Robinson ZIONISM Michael Stanislawski Available soon: KOREA Michael J. Seth TRIGONOMETRY Glen Van Brummelen THE SUN Philip Judge AERIAL WARFARE Frank Ledwidge RENEWABLE ENERGY Nick Jelley For more information visit our website www.oup.com/vsi/ Richard Earl Topology A Very Short Introduction Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Richard Earl 2019 The moral rights of the author have been asserted First edition published in 2019 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019949429 ISBN 978–0–19–883268–3 ebook ISBN 978–0–19–256899–1 Printed in Great Britain by Ashford Colour Press Ltd, Gosport, Hampshire Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. In memory of Dan Lunn. Friend, colleague, tutor. Contents Acknowledgements List of illustrations 1 What is topology? 2 Making surfaces 3 Thinking continuously 4 The plane and other spaces 5 Flavours of topology 6 Unknot or knot to be? Epilogue Appendix Historical timeline Further reading Index Acknowledgements Thanks go to Martin Galpin, Andy Krasun, Natalie Lane, Marc Lackenby, Kevin McGerty for their comments on and help with draft chapters. Thanks especially to Marc for his encouragement to write the book. List of illustrations 1 London underground maps (a) Archive PL / Alamy Stock Photo(b), © TfL from the London Transport Museum collection. 2 Examples of polyhedra 3 Manipulations of a cube to find its Euler number 4 The Platonic solids 5 Matt Parker and his football Matt Parker / YouTube. 6 Diagonals in a square must intersect 7 Two nonplanar graphs 8 Complicated Jordan curves (a) Get Drawings (b) David Eppstein/ Wikimedia Commons / Public Domain. 9 Original figure from Flatland showing how A Sphere is perceived by A Square The History Collection / Alamy Stock Photo. 10 The unknot and the trefoil 11 The torus 12 Making a cylinder 13 Making a torus 14 Gluing instructions for a triangle, pentagon, and square 15 Valid and invalid subdivisions of a sphere 16 Connected sums with tori 17 The Möbius strip (b) Dotted Yeti / Shutterstock.com. 18 Moving an oriented loop around a Möbius strip 19 The Klein bottle and projective plane 20 The real line and complex plane 21 Visualizing Riemann surfaces 22 Distance, speed, and acceleration on a journey 23 Examples of graphs 24 Continuous and discontinuous functions 25 The rigorous definition of continuous and discontinuous 26 The intermediate value theorem 27 The blancmange function 28 Visualizing different metrics 29 Graphs of functions of two variables 30 Open balls in the plane 31 Open sets in the plane 32 Visualizing convergence 33 Relating to a disconnected subspace 34 Relating to path connectedness 35 Examples of a minimum, maximum, and saddle point 36 Critical points of functions on a torus and sphere 37 Vector fields on the sphere and torus 38 Examples calculating indices of vector fields 39 Loops on a torus 40 Functions on the disc 41 The unknot and trefoils 42 The three Reidemeister moves 43 The granny and reef knots 44 Prime knots with seven or fewer crossings 45 The knot group of a trefoil 46 Links involved in the skein relation 47 Simple examples of links 48 Calculating the Alexander polynomial of a trefoil Chapter 1 What is topology? As you read this, passengers the world over are travelling on metro (or subway or underground) trains. There are around 60 billion individual journeys made annually on such metro systems. But whether this be in Tokyo, London, São Paolo, New York, Shanghai, Paris, Cairo, Moscow, those travellers are perusing maps for their journeys that are crucially different from maps in atlases or seen on geography classroom walls. Foremost in the minds of those passengers are the connections they need to make—getting out at the right station and changing to the correct new line. They are not interested in whether the map’s left–right lines do indeed run west–east, or whether they really did make a right angle turn when they changed lines, as depicted on the metro map. The oldest metro network in the world is the London Underground. When first produced, the underground maps superimposed the different train lines onto an actual (geographically accurate) map of London, as shown in Figure 1(a). A first version of the current map was designed by Harry Beck in 1931 as in Figure 1(b). Beck’s map, and the current underground map, are not wrong. Rather they transparently show information important to travellers—for example, the various connections between lines and the number of stops between stations. It is an early example of a topological map and demonstrates the different focus of topology—which is all about shape, connection, relative position—compared with that of geometry (or geography) which is about more rigid notions such as distance, angle, and area. 1. London underground maps (a) Geographically accurate 1908 map, (b) Beck’s topological 1931 map. Topology is now a major area of modern mathematics, so you may be surprised to learn that an appreciation of topology came late in the history of mathematics. The word topology—meaning ‘the study of place’—wasn’t even coined until 1836. (‘Geometry’ by comparison is an ancient Greek word and ‘algebra’ is an Arabic word, with its mathematical meaning dating back to the 9th century.) Just why this was the case is not a simple question to address, though we will see some aspects of topology developed as mathematicians sought to put their subject on a more rigorous footing. Topology is a highly visual subject that lends itself to an informal treatment and this book will give you a sense of topology’s ideas and its technical vocabulary. A topologist’s alphabet As a first example, to convey how differently topologists and geometers see objects, consider what capital letters a topologist would deem to be the ‘same’. Using the sans serif font, the four letters E F T Y are all topologically the same. They are not congruent, meaning that none of the letters can be picked up, and rotated or reflected, and then put down as one of the other letters. But I hope you can envisage, if allowed to bend, stretch, or shrink the letters, how any of them might be transformed into one of the others. To a topologist these four letters are homeomorphic to one another. The geometer would notice that the angle made by the arms of the Y is different from any angle found in the other letters. The topologist, on the other hand, would be happy to flatten the arms of the Y, and stretch its body a little, to give the T shape. Likewise the E could have its bottom rung bent around to the vertical, and then shortened somewhat, to make the F. Finally doing the same to the top of the F would make a T on its side. These four letters can be continuously deformed into one another and back again. Broadly speaking this is what it is to be homeomorphic, to be topologically the same. But what is it about these letters that makes them topologically different from other letters? Another collection of letters that are topologically the same as one another is: C G I J L M N S U V W Z. Topologically all of these are equivalent to a line segment and it’s not hard to imagine how each might be formed by bending and stretching a suitably mutable letter I. So hopefully you’re convinced that the letters in the second list are all homeomorphic to one another, but what makes this second collection topologically different from the first list? Note, for each letter E, F, T, Y, that every point lies on a distorted bit of line with one exception. In each of these letters there is a single point that might be described as a Tjunction. These Tjunctions are highlighted below. One way in which these Tjunctions are special is that, if removed, the remainder of the letter is disconnected into three parts; the removal of any other point would leave just two parts remaining. In whatever ways we might bend and deform an E the deformed version would still include a single Tjunction. As none of the second set C … Z has such a Tjunction then none of them can be a deformed version of an E (or an F, T, or Y ). This gives a genuine sense of how mathematicians resolve the question: are two shapes the same topologically? This either amounts to finding some means of continuously deforming one into the other, or involves finding some topological invariant of one that does not apply to the other. The word invariant is used in different contexts in mathematics: for example, if you shuffle a pack of cards, there will still remain fiftytwo cards afterwards and four suits, these are invariants; but the top card may have changed and the jack of clubs may no longer come before the eight of diamonds, and so such facts aren’t invariants of a shuffle. A topological invariant is something immutable about a shape, no matter how we stretch and deform it. In the above example we used the presence of a Tjunction as our topological invariant. You might note that an E includes four right angles whilst an F contains only three. The presence of four right angles is a geometric invariant and so shows that E and F are not congruent (i.e. not geometrically the same), but—working topologically—we are permitted to unbend these right angles and so right angles are not important from a topological point of view. Rather they’re mutable aspects of a shape and not topological invariants. The remaining twentysix letters, grouped topologically, break down as: DO, KX, AR, B, PQ, H. You might want to take a moment thinking about what makes an A different from a P or O different from Q. In fact, the O introduces an important topological invariant that separates it from both I and E. The shape of the O is different as it makes a loop. Technically O is not simply connected, a topic we will discuss more in Chapter 5. Euler’s formula One of the first topological results was due to Leonhard Euler (pronounced ‘oiler’), a titan of 18thcentury mathematics and one of the most prolific mathematicians ever; his formula dates to around 1750. The result relates—at first glance—to polyhedra, threedimensional objects such as cubes and pyramids (Figure 2). It is also so fundamental—a straightforward observation at least—that it is surprising ancient Greek mathematicians missed it. 2. Examples of polyhedra (a) A cube, (b) A Squarebased pyramid. Looking at the cube, we can see that it is made up of vertices (the corners of the cube—the singular is ‘vertex’), these vertices being connected by edges and that these edges then bound (square) faces. For the cube the number of vertices V equals 8, there are E = 12 edges and F = 6 faces. For the (squarebased) pyramid we have V = 5, E = 8, and F = 5. No pattern may be evident immediately but if we include the four other socalled Platonic solids—tetrahedron, octahedron, dodecahedron, icosahedron (Figure 4)—and other familiar polyhedra, we create Table 1. Table 1. Vertices, edges, faces for various polyhedra 4. The Platonic solids (a) Tetrahedron, (b) Cube, (c) Octahedron, (d) Dodecahedron, (e) Icosahedron. (We shall see soon that the truncated icosahedron is familiar to us—just not by that name!) For all the geometry that the ancient Greeks knew, it seems striking that this pattern eluded them, but we will prove now—or more honestly sketch a proof of—Euler’s formula which states, for a polyhedron with V vertices, E edges, and F faces, that In the proof our aim will be to begin with a polyhedron and manipulate it in certain ways—for example, we might remove or subdivide faces—but in all cases we will carefully track the effect our manipulation has (if any) on the number . If, after such manipulations, we arrive at a simplified situation where we know what equals, and we know the effects our manipulations had on that number, then we may be able to work backwards to find what was originally. We begin then with a polyhedron, and first remove one of the faces. This has the effect of reducing by 1 as F has decreased by 1. Now that the polyhedron has a missing face—effectively the polyhedron has been punctured—it can be flattened into the plane, taking care that all the vertices, edges, faces present on the punctured polyhedron remain and are connected in the plane in the same manner they were on the punctured polyhedron. For example, if we removed one face from a cube and flattened the remaining cube then we would have something like Figure 3(a). 3. Manipulations of a cube to find its Euler number (a) A flattened, punctured cube, (b) With flattened faces triangulated, (c) Removing a triangle, (d) Removing a triangle. The next manipulation is to subdivide each of the flattened faces into triangles—as has been done to the flattened cube in Figure 3(b). Introducing a single triangle has the effect of increasing F by 1—what was one face becomes split into two—of increasing E by 1—the new edge, introduced to make a triangle—and doesn’t change V. So there is no overall effect to as we keep introducing triangles; the increase of 1 to F, a term that is added in the formula, is precisely balanced by the increase in E which is a term we subtract. When this has been done for each flattened face (as in Figure 3(b)) then is still just one less than it was originally. We now remove the triangles one at a time. For example, if we remove the bottom triangle from Figure 3(b) to make Figure 3(c), then we remove one edge and one face and, by the same reasoning as before, this has no overall effect on . Similarly, we might then remove the rightmost triangle to create Figure 3(d), the manipulation again having no effect on . But the bottom triangle of Figure 3(d) is connected differently. If we remove that triangle then we remove 2 edges, 1 face, and 1 vertex. The algebra is a little more complicated this time, but again removing 1 vertex and 1 face means goes down 2 but this is countered by removing 2 edges as E is a term we subtract. Or if you prefer more formal algebraic reasoning, we are just saying Let’s summarize what’s happened so far: • We removed a face and decreased by 1. • We flattened the polyhedron into the plane—all V, E, F remain, so no change to . • We subdivided the flattened faces into triangles—this had no effect on . • We kept removing triangles from the edge of the flattened polygon—each removal having no effect on Eventually only a single triangle will remain, having removed all others. A triangle has a single face, three vertices, and three edges, so that equals . This is the value of that we finish with. The only manipulation that ever changed was that very first removal of a face which reduced it by one; initially then it was the case that This ‘sketch proof’ was given by AugustinLouis Cauchy in 1811. It’s worth highlighting there are several ‘i’s still to be dotted to make a proof with which a professional mathematician would be happy, but also noting how much of the idea of the proof is genuinely here. We didn’t take care describing how we removed triangles from the boundary of the flattened polygon; if we’d been careless we might have removed a triangle that disconnected the flattened polygon into two separate polygons, and we should have taken time to make sure such an occasion can always be avoided. Other issues will become more apparent in Chapter 2 but these ‘i’s can indeed be dotted. In Proofs and Refutations, the Hungarian philosopher Imre Lakatos used the specific example of Euler’s formula, and historical efforts to prove it, to highlight how hard it can sometimes be to generate a watertight proof and to also raise the question of when a theorem properly becomes part of mathematics or has mathematical content. It’s also worth mentioning that René Descartes had, over a century before Euler, demonstrated a theorem for polyhedra that is equivalent to Euler’s formula; his theorem was in terms of ‘angular defects’ at vertices. All of a sudden we are back in the geometrical world and it’s less than clear that there is a genuinely new subject, an importantly different way of mathematical thinking, that Euler’s fingertips were brushing against. Euler’s formulation encourages appreciation of the result as something a little new—in Euler’s terminology, rather than Descartes’s, it’s much clearer that the connection of the vertices, edges, and faces is what counts, but historically we are still a long time from a deeper appreciation of topology as a fundamental mode of mathematical thinking. There are five Platonic solids Platonic solids are polyhedra with regular faces that are all congruent (geometrically the same) and which meet in the same manner at each vertex. There are infinitely many regular polygons—equilateral triangles, squares, regular pentagons, etc.—but in 3D it turns out that there are just five regular solids which have been known since antiquity. These are shown in Figure 4 and with Euler’s formula we can show there are just these five. Consider a regular polyhedron with V vertices, E edges, and F faces. As the solid is regular then each face is bounded by the same number of edges; let’s call this number n. Likewise there is a common number m for how many edges meet at each vertex. So, with the cube, (the faces are squares) and (three edges meet at each vertex). Continuing with the cube as our example for now, think about how we can make a cube by gluing together the edges of six squares. We begin with 6 separate squares so that, before any gluing happens, there are 6 squares, 24 edges, and 24 vertices. Note that to make the cube it takes two ‘unglued’ edges to make each edge of the cube (which agrees with there being 24/2 = 12 edges) and it takes three ‘unglued’ vertices to make a single vertex of the cube (again there are 24/3 = 8 vertices). More generally, when we have F faces each with n edges, we would have nF edges before any gluing. It takes two of these unglued edges to make a single edge of the polyhedron which then has edges. There are as many unglued vertices as unglued edges, namely , and these will be glued together to make vertices on the polyhedron as it takes m unglued vertices to make one glued vertex on the solid. Putting these expressions for V and F into Euler’s formula we get (The equation has been rearranged to make F the subject of the equation so that .) We can then divide both sides of the above equation by 2E and rearrange to find As 1/E is positive this means that So m and n can’t both be very large as then and would be very small and their sum would not exceed . Also recall m and n are positive whole numbers, so there aren’t many options and it’s not hard to find all their possible values. It’s impossible for both m and n to exceed 4 as then would be less than . So either or or or (with perhaps more than one of these being true). For example, if , the only n for which the inequality is true are . If then equals or less and the inequality is not true. For the three cases of and we have A similar calculation for leads to , , and when we find , . In all these five cases we can use the previous formulas to work out the numbers of vertices and faces . We can put the full details into Table 2. Table 2. Possible m and n values for the Platonic solids If we are seeking to be rigorous here, we should really point out that the previous calculations show that there are at most five possible pairs of values that m, n can take. Those calculations limit the possibilities, but do not necessarily mean that there is a Platonic solid for each of these cases, nor preclude there being more than one Platonic solid for permitted m and n—it might be that there are two different Platonic solids with three pentagons meeting at each vertex. Listed in Table 2 are the five Platonic solids, and so we can see that there is at least one solid for permitted m, n. And it’s not hard to appreciate why there can be at most one. In the case where , then three squares meet at each vertex; seemingly this only tells us something about parts of the solid, but if we follow this recipe of attaching three squares at each vertex then there is only one way to progress building up the solid—it’s not clear that this recipe will actually lead to a complete solid, but it does show that there can be at most one Platonic solid for each allowed m, n. (As an aside, you may have noticed that and , and , are solutions if we permit E to be infinite. These ‘solutions’ correspond to tessellations of the plane where four squares meet at a vertex, where three regular hexagons meet at a vertex (as with honeycombs), and where six equilateral triangles meet at a vertex. There are also some patterns apparent in Table 2 for the values of V, E, F for the cube and octahedron, and likewise the dodecahedron and icosahedron. This is because these solids are dual to one another—this means that the midpoints of the faces of a cube make an octahedron, and vice versa; likewise the dodecahedron is dual to the icosahedron and the tetrahedron is selfdual in this sense.) Footballs In 2017 the mathematics popularizer Matt Parker began a petition seeking to get road signs to football stadia corrected. You may not have noticed the inaccuracy of such signs in the past, perhaps being happy just to know you’re travelling the right way for the game. But it’s clear (Figure 5) that the sign’s football does not resemble the actual football that Matt is carrying. A football’s surface is made from pentagons and hexagons and the everyday football is more formally known as a truncated icosahedron. (It can be created from an icosahedron by planing down, around each vertex, the five edges meeting there. If we plane down onethird of each of those five edges, we create a new pentagonal face and continued planing eventually shrinks all the triangular faces to hexagons.) I think though the irksome principle for Matt was not that the sign’s football was badly drawn, it was in fact impossibly drawn. 5. Matt Parker and his football. There is no way that a sphere can be made by stitching together hexagons as shown on the sign. That would be an example where and , using the previous notation, and whilst we can cover the plane in this way—which may be why the sign looks plausible at first glance—making a football this way is mathematically impossible. In fact, Euler’s formula shows us how many pentagons and hexagons there are on a football. Recalling how to truncate an icosahedron we see there are as many pentagons as original vertices (12) and hexagons as original faces (20), but Euler’s formula can show this is the only way to construct such a football. Say a football has P pentagonal faces and H hexagonal faces. Then, before gluing these together, we have 5P + 6H unglued edges and the same number of unglued vertices. Looking at Matt’s ball we can see that (i) two unglued edges are needed to make an edge on the football and (ii) three unglued vertices make a vertex with (iii) two hexagons and one pentagon meeting at a vertex; from (iii) we see there are twice as many ‘unglued’ vertices collectively on the hexagons as on the pentagons. So If we put these values into Euler’s formula we find that which simplifies and rearranges to P =12 and the equation 10P = 6H yields H =20. This, then, is the only way to make a football if we follow the rules (i), (ii), (iii). Graph theory Let’s change tack a little and consider the following problem. The square PQRS in Figure 6 has diagonally opposite vertices P and R, Q and S. If we were to draw curves from P to R, and from Q to S, curves which remain within the square as in Figure 6, then surely those curves would have to cross at least once. (In Figure 6 there are three intersections.) This seems obvious—and is true—but how would you go about proving this? 6. Diagonals in a square must intersect Curves PR and QS in the square PQRS. Before more is said, it might be worth stressing how characteristic of a topological question this is. The curves PR and QS need to connect their end points. Those curves don’t need to be polygonal, or have welldefined gradients, or be defined by specific functions. They need to connect the end points in some continuous sense—fuller details in Chapter 3—but they are otherwise general paths from P to R and from Q to S that remain in the square. At first glance, this problem might seem quite removed from the polyhedra we were just discussing. However, Figure 6 doesn’t look that different from Figures 3(a)–(d). We have vertices (P, Q, R, S and any points where the curves PR and QS meet), edges running between these vertices (though admittedly they’re now curved), and we have faces bounded by those edges. It was crucial to the proof of Euler’s formula that for each of Figures 3(a)–(d). If we also include the outside region as a face—essentially the one removed so we could flatten the polyhedron—then we arrive back at Euler’s formula . (By this reckoning , , in Figure 6.) So suppose, somehow, we could draw curves PR and QS in the square PQRS which don’t intersect. We’d then find • the four corners P, Q, R, S. • the square’s four sides and the curves PR, QS. • the outside of the square, above PS, below QR, right of PQ, left of SR. But this leaves us with and so such a scenario is impossible by Euler’s formula. Graph theory is an area of mathematics that models networks in a wide sense: physical, biological, and social systems, variously representing transport networks, computer networks, website structure, evolution of words across languages and time in philology, migrations in biology, etc. A graph is a collection of points called vertices, with these vertices connected by edges. We will also assume that graphs are connected, meaning that there is a walk between any two vertices along the edges. This definition may be extended to include oneway edges—directed graphs or digraphs—and weights might be introduced to edges representing the difficulty—in terms of time, distance, or cost—of travelling along a particular edge. Some graphs are planar, meaning that they can be drawn in the plane without their edges crossing (at points that aren’t vertices). The two graphs K5 and K3,3 in Figure 7 are importantly not planar. The complete graph on 5 vertices, denoted K5, has a single edge between each pair of the 5 vertices making 10 edges. You might think that K5 is planar as it’s drawn in Figure 7(a)—the point is that, so drawn, many of the edges’ crossings don’t occur at vertices and to deem these crossings as vertices would mean we were no longer considering K5 which has only 5 vertices. If we could properly draw K5 in the plane there would be 10 triangular faces v1v2v3, v1v2v4, through to v3v4v5. We’d then have that equals and so K5 is not planar. 7. Two nonplanar graphs (a) The complete graph K5, (b) The complete bipartite graph K3,3. K3,3 is the complete bipartite graph between two trios of vertices. A somewhat subtler argument shows K3,3 is not planar. Note that K3,3 has vertices and edges. If drawn in the plane this would mean . But a face of K3,3 would have at least four edges as its perimeter necessarily runs from a v to a w to a different v and to a w and only then may return to the original v. So, counting the edges by going around all the faces, we would get a total of at least edges. However, as an edge can bound at most two faces this would mean we’d have at least edges, which is our required contradiction. The Polish mathematician Kazimierz Kuratowski proved in 1930 that a graph is planar precisely when neither a copy of K5 nor K3,3 can be found within the graph. We will in due course see that K3,3 can be drawn on other surfaces such as a torus (Figure 13(c)). Nasty surprises Euler arrived at his formula a century before the word topology was coined. His formula is characteristic of a visual side of topology naturally aligned with geometry. But topology, as a subject, would develop along various themes and in particular had an important role in the foundational work mathematicians were doing around the start of the 20th century. As I hinted earlier, topology’s rise may have been hampered by a traditional mindset that some of its questions had obvious answers. For example Camille Jordan, as late as the 19th century, proved the following: a curve in the plane, which does not cross itself and which finishes back where it began—a curve which we would now call a Jordan curve—splits the plane into two regions, the technical phrase for these regions being connected components. One of these regions is bounded, the inside, and the other is unbounded, the outside, and this is the Jordan curve theorem. Earlier mathematicians would have happily thought this obvious and the first rigorous proof didn’t appear until 1887. You may agree with those earlier mathematicians that the result can be safely assumed. Maybe even the Pollocklike Jordan curve in Figure 8(a) does not sway your view of the intuitiveness of the result. 8. Complicated Jordan curves (a) A more complicated Jordan curve, (b) A Jordan curve with positive area. In Figure 8(b) is the Knopp–Osgood curve which, for all its fractallike appearance, is a Jordan curve. Astonishingly it has a positive area—that is the curve itself has positive area, we’re not referring to some region that it bounds. Would you have said a moment ago that it’s obvious that curves can’t themselves have area? You shouldn’t worry too much in the sense that most things those early mathematicians thought to be true turned out to be true, once properly understood and qualified, but mathematicians towards the end of the 19th century were getting nervous about the rules and assumptions that mathematics relied on. A related problem within topology at that time was rigorously defining what dimension means. Again this had previously been treated as an intuitive concept, only for mathematicians to begin finding spacefilling curves that pass through every point in the plane or other weirdandwonderful spaces that can reasonably be assigned dimensions that are not whole numbers—spaces that would now be called fractals. An early theme of topology was this general topology or pointset topology seeking to address what it means to be a set, to be a space, etc. Metric and topological spaces were introduced—to be discussed in Chapter 4—each being attempts to describe general structures where continuity could be defined. Set theory deals with collections that are essentially just thingsinabag. This general topology sought to define ways in which objects might be considered ‘close’ to one another, with the aim being to define continuity in a broad setting. A Flatland mindset The novella Flatland: A Romance of Many Dimensions, written in 1884 by Edwin Abbott, is a satire on Victorian mores. The narrator is ‘A Square’, an inhabitant of Flatland, a planar world having just two dimensions. The culture of Flatland and the logistics of living in two dimensions are fully described, implicitly highlighting some of the narrowmindedness of Victorian culture—for example, women are one rather than twodimensional beings. The story doesn’t explicitly discuss topology, but in its description of worlds with different dimensions and implications for the inhabitants, it provides a useful metaphor for understanding certain aspects of topology. For example, A Square is visited at one point by A Sphere. Being a threedimensional object, A Sphere can only be perceived by Flatlanders as a circular crosssection (Figure 9). By moving up and down—relative to Flatland’s plane—A Sphere can grow, shrink, and even disappear entirely. In a similar manner, to truly understand the topology of a space, we have to begin thinking like inhabitants of that space. 9. Original figure from Flatland showing how A Sphere is perceived by A Square. Topology is often characterized as rubbersheet geometry. It’s a somewhat clichéd metaphor, but it’s also slightly inaccurate. It gives a correct sense of topology being more about shape and less rigid than geometry in its focus. On the other hand, in Chapter 6 we discuss knots, and as a (genuine) knot—like the trefoil—and the (unknotted) circle (Figure 10) cannot be continuously deformed into one another in 3D then you might be tempted to say the circle and trefoil are not homeomorphic, but they are. 10. The unknot and the trefoil (a) The unknot, (b) The trefoil knot. The knottedness of the trefoil says something about its position in 3D. In fact, all knots are homeomorphic to a circle. To better appreciate this, you might imagine life as an ant living on either the circle or trefoil. As the ant moves around either the unknot or trefoil it has a sense of being on a loop, but the ant has no notion of whether it is living on a knot. It is only by being able to view things from outside the two loops, and looking on from a position in the ambient space, that we are able to recognize one loop as knotted as compared with the other. This Flatland mindset will prove useful again later when we meet subspaces. Topology would advance on various fronts in the 19th and 20th centuries. In particular, Bernhard Riemann would early on show the usefulness of a ‘topological mindset’, introducing Riemann surfaces into the study of polynomial equations and demonstrating some deep connections between topology and many other areas of mathematics. Chapter 2 Making surfaces The shape of surfaces Recall Euler’s formula states for a polyhedron. Various details of the proof were brushed under the carpet, the most significant of these being the claim that, once a face is removed from a polyhedron, the remaining polyhedron can be flattened into the plane. This was true for the polyhedra we were considering, but the claim says something important about the shape of the remaining polyhedron that was perhaps unintentional. In any case, the next example will either make us question what we mean by a polyhedron or have us looking to generalize Euler’s formula. For Figure 11(a)’s ‘polyhedron’, a count of vertices, edges, and faces shows that , , , giving which seems to disprove Euler’s formula. We are left with a few alternatives: either the object in Figure 11(a) should not be considered a polyhedron, or we need to restrict Euler’s formula to a certain type of polyhedron, or we need to adapt and generalize Euler’s formula into a version that remains true for a broader family of polyhedra. 11. The torus (a) A polyhedron with one hole, (b) A torus. The most obvious issue with this new ‘polyhedron’ is the hole through its middle. This is not immediately reason enough to exclude it as a polyhedron, but this shape, once a face is removed, does not leave a remainder that can be flattened into the plane, making our earlier proof invalid. We need either to restrict Euler’s formula to polyhedra without holes, or we need to work out the correct values for polyhedra with holes. Recalling the rubbery nature of topology, we might recognize that the polyhedra of Chapter 1 all had the same underlying spherical shape. If allowed to smooth out those polyhedra—the pointy vertices and the ridgy edges—we could transform each of those polyhedra to a sphere, covered with a patchwork of curved faces, just like Matt Parker’s football (Figure 5) was covered with curved pentagons and hexagons. But however we smooth down our new polyhedron we can’t make a sphere, rather we would make a torus, the shape of a doughnut with a hole through it (Figure 11(b)). Perhaps then all of the examples of Chapter 1—including the proof of Euler’s formula—point to the Euler number of the sphere being 2. And Figure 11(a) is a first example suggesting the Euler number of the torus is 0; this would mean the number equals 0 however we divide up the torus. All this could become quite involved unless we have a way of efficiently describing surfaces—including more complicated ones than the sphere or torus—and for systematically calculating their Euler numbers, that is the value common to all surfaces of a certain underlying shape. Gluing surfaces together A useful way of constructing surfaces is to begin with a polygon and pairwise glue together the edges of the polygon, the way a model kit might direct you to ‘glue tab A to tab B’. How might we make a torus in this manner? If we begin with a (suitably elastic) square (Figure 12(a)), bend it around (Figure 12(b)), and glue the opposite edges e1 and e3 so that the vertices v1, v2 get glued respectively to v4, v3, and likewise for all other opposite points of e1 and e3—as signified by the two arrows—then we will make the cylinder drawn in Figure 12(c). Note that the edges e2 and e4 on the original square have become the two circular ends of this cylinder. We can then glue together these circular ends; if we do this so that the opposite points of the original e2 and e4 are glued together, then we make a torus as in Figure 13(a). 12. Making a cylinder (a) A square with identified edges, (b) Making the cylinder, (c) A cylinder. 13. Making a torus, (a) Torus with v and e1, e2 drawn, (b) Square with gluing instructions, (c) K3,3 on the torus. Note that the four corners of the original square—denoted v1, v2, v3, v4—have all been glued together to make a single point v on the torus. Similarly, the edges e1 and e3 have been glued together to form a circle going around the outside of the torus and e2 and e4 have been glued together to form a different circle going through the hole of the torus, these two circles meeting at the point v. Importantly, the torus’s shape is fully described by a square with directions for how the edges are to be glued together. Mathematicians would draw this squarewithgluinginstructions as shown in Figure 13(b). The single arrows—and importantly the directions in which they’re drawn—show how those two edges are glued, and the double arrows (again noting directions) tell us how the other pair of edges is glued. As an aside, the graph K3,3, which we saw can’t be drawn in the plane (Figure 7(b)), can be drawn on the torus (Figure 13(c)). The reason K3,3 can be drawn on the torus is because a torus has a lower Euler number of 0 and, arguing as before, it can then be shown that F = 3. If you look carefully at Figure 13(c) you will see that there are indeed three faces (quadrilaterals v1w2v2w1 and v3w3v2w2 and a single hexagonal face v1w2v3w1v2w3). How does all this help with determining Euler numbers? With a single square and gluing directions, we have been able to make an object with the shape of a torus and this is certainly a conciser description of a torus than Figure 11(a) for which , , . But is this glued square enough to work out the Euler number of a torus? The answer is yes if we’re careful when thinking about just how the original vertices and edges glue together. On the original square there was just one face—the square’s interior—four unglued edges (labelled e1, e2, e3, e4), and four unglued vertices (labelled v1, v2, v3, v4). However, once we’ve followed the gluing directions, those four edges have become the two circular edges on the torus and the four vertices have become one point v, as in Figure 13(a). And there is still one ‘face’ on the torus—the square’s interior has been stretched to become all of the torus except those circles and v. So when we use this glued square to calculate the Euler number of the torus we get the answer which agrees with our calculation from Figure 11(a). If you prefer Figure 13(a) showing the torus with the four glued vertices becoming one, and the four edges become two loops on the torus, then Figure 13(b) will only appear as an unfinished DIY job. But the single surface obtained from gluing the triangle, pentagon, and square in Figure 14, following all the gluing directions a, b, … f according to the arrows, might reasonably start stretching your visualization skills. But we can still work out the Euler number of this chimera and seek to understand just what surface we are looking at. This time there are three faces (the triangle, pentagon, and square) and the twelve unglued edges of the polygons make six glued edges a, b, c, … f on the surface. How many vertices will we ultimately have? There were twelve unglued vertices originally but various of these get glued together as we make the surface. For example, v1 and v5 are glued together as they are both at the rear end of the edge marked a. In fact, we can chase around these gluings to see just how many vertices we have: 14. Gluing instructions for a triangle, pentagon, and square. v1 and v5 are glued (rear end of a) v5 and v9 are glued (rear end of f ) v9 and v4 are glued (rear end of d) v4 and v12 are glued (front end of f ) v12 and v2 are glued (rear end of c) v2 and v7 are glued (rear end of b) v7 and v11 are glued (front end of e) v11 and v1 are glued (front end of c) So eight different (unglued) vertices v1, v2, v4, v5, v7, v9, v11, v12 all get glued together as a single vertex on the surface. In a similar fashion we can see that the remaining four vertices v3, v6, v10, v8 get glued together (in that order). So once made, the surface has 2 vertices, 6 edges, and 3 faces giving an Euler number of . Just what surface have we made? Getting the right answer: subdivisions We need now to make clear just what surfaces we are considering—closed surfaces—and how they can be divided up into vertices, edges, and faces. A closed surface is one without a boundary, such as a torus or sphere, but not the cylinder of Figure 12(c). Our process of making a torus begins with a square and at that point our surface has a boundary consisting of its four edges; when we glue two edges to make a cylinder then the surface still has a boundary, namely its top and bottom circles. Once the torus has been made, no boundary points remain unglued. Secondly, we can only calculate the correct Euler number of a closed surface if we are careful dividing it up. Cubes, footballs, dodecahedra, and pyramids are all valid ways of ‘subdividing’ the sphere into vertices, edges, and faces. In each of these cases we obtained an Euler number of 2. However here are other ways we might subdivide the sphere that seemingly produce differing Euler numbers (Figure 15). 15. Valid and invalid subdivisions of a sphere (a) V = 0, E = 0, F = 1, (b) V = 0, E = 1, F = 2, (c) V = 1, E = 0, F = 1, (d) V = 1, E = 1, F = 2, (e) V = 1, E = 2, F = 3, (f ) V = 2, E = 0, F = 1. In order the values of for the six spheres in Figure 15 are 1, 1, 2, 2, 2, 3 and we know the correct Euler number equals 2. So any old subdivision will not lead to a correct calculation of the sphere’s Euler number. For a collection of vertices, edges, and faces to make a permissible subdivision the following must be true: • an edge must start and finish in a vertex; • when two edges meet, they must meet in a vertex; • faces must be (distorted) polygons. Looking at these socalled six subdivisions, only two of these are in fact permissible, 15(c) and 15(d). In 15(a), 15(e), 15(f), there is a face that is not a distorted polygon; neither the whole sphere (15(a)) nor the punctured cummerbund (15(e)) nor the twicepunctured sphere (15(f)) are topologically the same as a polygon and so not permissible faces. In 15(b) and 15(e), the edges do not begin and end in a vertex. 15(e) is deliberately given to show that the correct Euler number can be incorrectly calculated. Looking back at the torus in Figure 13(a) and the surface in Figure 14, we calculated their Euler numbers using subdivisions consistent with the above three rules. Therefore, we correctly calculated their Euler numbers as 0 and –1 respectively. Connected sums Given two closed surfaces S1 and S2, then we can create their connected sum S1#S2. This is a way to glue surfaces together and a useful means of making new surfaces from the few we have so far met. Say S1 and S2 each has a subdivision that includes a ‘triangular’ face bounded by three edges. (The inverted commas here hint that, this being topology, the faces may not be that recognizably triangular in terms of having straight edges.) The connected sum S1#S2 is then created by removing these two triangular faces, so making two holes in the surfaces, and then gluing the two surfaces together along the boundaries of the holes, pairing up the three vertices and three edges with those on the boundary of the second removed face as, for example, in Figure 16. 16. Connected sums with tori (a) One torus with a “triangle” missing, (b) A torus with two holes as a connected sum. Helpfully there is a formula for the Euler number of S1#S2. In making the connected sum, we remove two triangular faces, the six different vertices on these triangles are glued to make three vertices on the connected sum, and likewise six edges are glued to make three. So the total number of faces has gone down by 2 and the total numbers of edges and vertices have each gone down by 3. As V and F are added in the formula for the Euler number, and E is subtracted, overall we have Or, if you prefer a more careful algebraic proof, say the original subdivision of S1 has V1 vertices, E1 edges, and F1 faces and define V2, E2, F2 similarly for S2. The number of vertices V#, edges E#, and faces F# on the connected sum is given by Finally Thinking in terms of connected sums helps us work out the Euler numbers of some more complicated surfaces. We know that a torus has an Euler number of 0. The connected sum is a torus with two holes (Figure 16(b)) and we see and similarly the torus with three holes, , has Euler number In fact, we can see that every time we make a connected sum with the surface gains one more hole and the Euler number reduces by 2. So the torus with g holes—which can be considered as , the connected sum of g copies of the torus —has Euler number The number g of holes in the surface is called the genus of the surface. Onesided surfaces At this point, we still can’t identify the peculiar surface from Figure 14 which has an Euler number of –1. So far we’ve only constructed surfaces with even Euler numbers and –1 is odd. In fact, with the tori , we’ve only met half the story and half of the closed surfaces. Recall how in Figure 12 we made a cylinder by gluing two edges of a square. We could, instead, have glued those two sides using reversed arrows (Figure 17(a)), introducing a single twist. So the points near v2 on e1 are glued to the points near v4 on e3 and those near v1 on e1 are glued to the points near v3 on e3. This would have created a Möbius strip, named after August Möbius who discovered it in 1858. 17. The Möbius strip (a) A square with identified edges, (b) Runners on a Möbius strip. The Möbius strip is unusual in only having one side—this is apparent in Figure 17(b) as the runners cover the entirety of the strip rather than just one side of it as they would if running around just the outside (or inside) of a cylinder. Or you can imagine painting the outside of a cylinder black and the inside white, but should you begin painting a Möbius strip one colour you would find yourself covering the entire strip in that colour. The Möbius strip is an example of a nonorientable surface. Like the cylinder it is a surface with boundary, but note its boundary is a single circle rather than two separate ones as with the cylinder. In Figures 18(a)–(d) we see an oriented loop—here a circle—moving around a Möbius strip. By an oriented loop I mean a loop with a given sense of direction, here initially (18(a)) appearing as clockwise to the reader. But as this loop moves around the strip (or equivalently moves left in the square) we see that when the circle returns to its original position (18(d)) that sense has now reversed and appears anticlockwise. If you are having a little trouble visualizing what’s happening to the loop, note in 18(b) and 18(c) how the points labelled P are glued together and likewise the Qs. In 18(b) most of the loop (on the left) looks to be clockwise running from P to Q, but as the loop appears on the right and continues from Q to P that sense is beginning to appear as anticlockwise. 18. Moving an oriented loop around a Möbius strip. Any surface on which it is possible to reverse the sense of an oriented loop is called nonorientable. If it is impossible to reverse a loop’s sense, then the surface is called orientable. Any surface that contains a Möbius strip is nonorientable as we could just send an oriented loop once around that strip to reverse its sense. A surface with an inside and an outside is orientable. To appreciate this, imagine walking around the outside of such a surface. Looking down to your feet on the surface you could draw a circle in a clockwise manner. As you wander around the outside of the surface you can consistently take your notion of clockwise across the whole surface. This means, in particular, that the tori , which we met earlier and which each have an inside and outside, are all examples of orientable surfaces. Returning to Figure 17(a), a partly glued square making a Möbius strip, there remain two unglued edges e2 and e4. We could glue these together as in Figure 19(a), but what surface would we make? Certainly a nonorientable one as it contains a Möbius strip (the shaded region). If instead we make this surface by gluing e2 and e4 first, we first create a cylinder with e1 and e3 as its circular ends. But to complete the surface, rather than bringing those circular ends together as with a torus, one circular end has to be glued backwards on to the other circular end—this is because of the reverse arrows on e1 and e3. Figure 19(b) shows how we might try to do this; we could take one circular end back into the cylinder and glue it to the other end from inside, and this way the reverse arrows line up properly. The surface made is called a Klein bottle, after Felix Klein who first described it in 1882. Being nonorientable, the Klein bottle does not have an inside and outside. 19. The Klein bottle and projective plane (a) A Klein bottle, (b) 3D depiction of a Klein Bottle, (c) A projective plane. There is a subtle problem with the Klein bottle in Figure 19(b). When we take the cylinder back into itself, some single points in space actually represent two distinct points on the Klein bottle. So this image is not a proper representation or embedding of the Klein bottle in 3D. In fact, it is impossible to construct a Klein bottle in 3D without such selfintersections as occur where the cylinder cuts back into itself. The relevant result demonstrating this impossibility can be viewed as a generalization of the Jordan curve theorem. That theorem concerned embedding circles in the plane with a Jordan curve having an inside and an outside. In a like manner when a closed surface is embedded in 3D, the surface again divides the remaining space into an inside and an outside and so the closed surface must be orientable. As the Klein bottle is nonorientable, it cannot be embedded in 3D. However, the Klein bottle can be embedded in 4D and this isn’t too hard to imagine if we treat the fourth dimension as time. The Klein bottle is twodimensional (as surfaces are) and so from this 4D viewpoint it is important to consider the Klein bottle as only existing for an instant, a certain ‘now’; for it to have a past or future would give it a third dimension. So when faced with bringing the cylinder back into itself—which would normally cause selfintersections—we can instead move that bit of cylinder gradually into the future (the fourth dimension), where the remainder of the Klein bottle doesn’t exist and then, once the cylinder has passed through the space its present self occupies, we can gradually bring that bit of the cylinder back into the present. The selfintersections no longer occur, as the distinct points of the Klein bottle that became merged in Figure 19(b) instead sit in the same point of space but crucially at different times. We can also determine the Euler number of the Klein bottle, again being careful to note how edges and vertices are glued together. The square is our only face; e1 and e3 are glued together, as are e2 and e4, making two rather than four edges; finally v1 is glued to v2 which is glued to v4 which is glued to v3 and so we have just one vertex, giving , the Euler number of the Klein bottle. Unfortunately, 0 is also the Euler number of the torus, so any hope we might have had that the Euler number alone is information enough to recognize the shape of a surface was simplistic. The torus and Klein bottle are different surfaces—the former is orientable (twosided), the latter not—and yet they both have the same Euler number. Another important nonorientable surface, which can be formed from gluing a square’s edges together, is the projective plane ℙ. In Figure 19(c) we assign e2 and e4 reverse arrows (in contrast to 19(a)). The surface formed is nonorientable, as it again contains a Möbius strip (the shaded region), and we can calculate the Euler number as before: again and but this time v1 and v3 are glued together and separately v2 and v4 are glued, so that . Hence ℙ has Euler number . The classification theorem Classification is an important theme in mathematics. A mathematical theory often begins with definitions and rules about certain mathematical objects or structures (say functions or curves) and seeks to prove results about them using those rules. It’s natural to search for examples satisfying those rules, preferably producing a complete list or classification of such objects. We are now close to classifying closed surfaces. Explicitly, we are seeking to give a complete list of all the closed surfaces, so that every closed surface is homeomorphic to (i.e. topologically the same as) one of the surfaces on the list, and the list contains no duplicates—each surface on the list can be shown to be topologically different from all others on the list. It turns out that the Euler number goes a long way to separating out the different surfaces, but we have seen that this cannot be the whole story as the torus and Klein bottle have the same Euler number whilst being different surfaces—the first is orientable, the second not. The only missing ingredient in the classification is that notion of orientability. So the first half of the classification theorem for twosided surfaces states: • An orientable closed surface is homeomorphic to precisely one of the tori where These tori are not topologically the same as one another as they have different Euler numbers—the Euler number of is 2–2g. A similar result holds for onesided closed surfaces. Just as the torus is a building block for the orientable surfaces, so can the projective plane ℙ be used to make the nonorientable surfaces. Recall that the projective plane ℙ has Euler number 1. So the connected sums ℙ#ℙ and ℙ#ℙ#ℙ have and more generally k copies of ℙ in a connected sum, a surface denoted ℙ#k, has Euler number 2–k. And the second half of the classification theorem for onesided surfaces states: • A nonorientable closed surface is homeomorphic to precisely one of ℙ#k where These surfaces are not topologically the same as they have different Euler numbers—the Euler number of ℙ#k is 2–k. Making a connected sum with ℙ is equivalent to sewing a Möbius strip into the surface. ℙ itself can be made by introducing a Möbius strip into a sphere; to do this we might make a tear in the sphere and then, rather than gluing the tear back together, we could instead assign reverse arrows to the two sides of the tear, thus introducing a Möbius strip. So the surface ℙ#k can be thought of as a sphere with k Möbius strips sewed in. Overall then, the classification theorem says that if we know the Euler number of a closed surface and whether it is one or twosided, then we know its topological shape. If you were wondering, where the Klein bottle is on this list, we know its Euler number to be 0 and we know it to be onesided. The only surface in the classification matching these facts is the surface ℙ#ℙ and this is topologically the same as the Klein bottle. We might create a yet more complicated connected sum such as which at first glance is not on our list. This surface is onesided and its Euler number equals so topologically it’s the same surface as ℙ#7. And at long last we are able to identify the surface we formed in Figure 14. That surface had Euler number –1 and so the surface is ℙ#ℙ#ℙ, this being the only surface on our list with that Euler number. Complex numbers Surfaces are a natural twodimensional extension of onedimensional curves which mathematicians had long been interested in but, historically, surfaces and their topology became of particular importance because of the work of 19thcentury mathematicians, most notably Bernhard Riemann. To understand Riemann’s motivation for studying surfaces, we need to take a brief foray into the world of complex numbers. Complex numbers have, at first glance, nothing to do with topology, but the need to introduce them here is a consequence of the deep interconnectedness of mathematics. In the mid19th century mathematicians found worthwhile reasons to think about older mathematics in new topological ways. It might then seem as though topology was somehow born of practical necessity for addressing these older problems. However I’d like to suggest a rosier picture of how mathematicians think: nothing will put a glint in the eyes of a generation of mathematicians, an itch to be thinking hard about the essence of mathematics, so much as a sense of there being something profound just around the corner and a deeper understanding of their subject tantalizingly beyond their fingertips. And so it was to prove. These socalled ‘complex’ numbers arose—somewhat uncertainly—from the work of Italian mathematicians during the Renaissance. For a long time mathematicians had been interested in the solutions of polynomial equations. These are equations involving powers and multiples of an unknown quantity, say x, such as This is a degree 3 equation, that being the highest power of x. A solution of an equation is a value of x which makes both sides equal. We can see that solves this equation because You might check that is a solution and so is . And that’s all of them! Three solutions . Other polynomials, though, seem to have no solutions. For example, the degree 2 equation has no real numbers as solutions. If you take a positive number x then its square x2 is also positive (and so cannot equal –1); if you take a negative number then its square is also positive; finally . So there are no solutions. If you prefer a more pictorial approach then you might draw the graphs of and , and the fact that these graphs don’t meet (Figure 20(a)) is again another way of showing that no number x solves the equation . Basically the problem is that negative numbers don’t have real square roots. 20. The real line and complex plane (a) Graphs of y = x2 and y = –1, (b) The real line, (c) The complex plane. And there the story might have ended except those Renaissance mathematicians found good reasons to ‘imagine’ that does have solutions, denoting a solution as i. This may seem somewhat ludicrous at first, but around 1530 a method was found for solving degree 3 equations. One problem was that this method necessitated calculations with square roots of negative numbers, even when all the equation’s solutions were real numbers. The worth of the number i became truly apparent with the proof of the fundamental theorem of algebra in 1799 by Carl Gauss. This theorem shows all the solutions of any polynomial equation have the form where a and b are real numbers. For example, the number solves the equation as shown by the calculation Numbers of this form, where a and b are real numbers and , are called complex numbers and the fundamental theorem of algebra says that a polynomial of degree n has (counting possible repeats) n solutions amongst the complex numbers. In the same way that real numbers are commonly represented on the real line (Figure 20(b)) the complex numbers can be represented as a plane, the complex plane (Figure 20(c)). A complex number such as can then naturally be identified with the point (1, 2) as shown. The real numbers occupy the horizontal axis—denoted ‘Re’—and the vertical axis ‘Im’ is called the imaginary axis. Complex numbers have a rich theory of their own which, for mathematicians at least, is reason enough to warrant their study. You may, though, be surprised to find that quantum theory, the physical theory that successfully models subatomic physics, is naturally described using the language of complex numbers and so physicists, chemists, and engineers all need to be well versed in the use of complex numbers. Riemann surfaces The introduction of complex numbers led to a much richer theory connecting algebra and geometry. In Figure 20(a) we see that the curves and don’t meet; if they did meet at a point (x,y) in the real xyplane then we’d have and no such x exists. But using complex numbers they do intersect at two points, at and at . The fact that a degree 2 curve and a degree 1 curve meet in points in this case is not entirely coincidental. More generally it is the case that, if properly counted, a degree m curve and a degree n curve intersect in points. Multiple contacts need to be counted properly—so a line tangentially meeting the curve would count as a double contact, so that there are still intersections. The final finesse, when counting intersections, is to include points at infinity. For example, two parallel lines—each degree 1 curves—are still deemed to meet at a point at infinity so that there is intersection as expected. Using real numbers, the graph of is a onedimensional curve lying in the twodimensional xyplane. In this case x and y are everyday real numbers and the curve consists of all points (x, x2) where x is a real number. We might instead consider the same equation where x and y can now be complex numbers. Again all the points satisfying are of the form (x, x2) but this time x can be any complex number. When using real numbers, the input x represents some point of the xaxis and the corresponding output x2 can be plotted distance x2 above the point (x, 0) in the xyplane (Figure 20(a)). However, when it comes to using complex numbers, the input is itself twodimensional. The x‘axis’ is a version of the complex plane, the y‘axis’ a second version, and the complex xy‘plane’ is in fact fourdimensional. ‘Above’ the point (x, 0) is a point (x, x2), and together the points (x, x2) make a twodimensional surface situated in the fourdimensional complex xyspace. All the points such as (2, 4) that were on the original real curve are still present, and make up a crosssection of the complex surface; present too now are points like (i, –1) and (2+i, 3+4i). If we separate out these complex numbers into their real and imaginary dimensions, then we might instead represent these points as and their fourdimensional nature is a little clearer. The curve sits in the real xyplane as a curved version of the xaxis (Figure 20(a)); the curve and axis are topologically the same with a homeomorphism between the two just pushing each point (x, 0) up to the point (x, x2). If we include also the curve’s point at infinity bringing together the curve’s ‘ends’ then the curve topologically becomes a circle. When using complex numbers, the curve sits in complex xyspace as a curved version of the xaxis which, remember, is itself a twodimensional complex plane. When we include the point at infinity this brings together this curved plane as a sphere. (This is the reverse process of puncturing a sphere to get the plane that we met earlier in the proof of Euler’s formula.) So, the complex version of , if we include its point at infinity, is topologically a sphere, a surface; this is called the Riemann surface of . We might similarly consider the Riemann surfaces of higher degree equations. In Figure 21(a) we have the real crosssection described by the degree 3 equation . On the left is a loop, and when we add the point at infinity to the curve on the right then topologically this real crosssection becomes two loops; so it might not be surprising that the whole complex version, the Riemann surface, in this case is a torus with Figure 21(a) just being a crosssection of that torus. In Figure 21(b) we have a real crosssection with a singular point where the curve crosses itself. Topologically the complex version of this curve is a pinched torus as in Figure 21(c). 21. Visualizing Riemann surfaces (a) Graph of y2 = x(x – 1)(x – 2), (b) Graph of y2 = x(x – 1)2, (c) A pinched torus. Provided there are no singular points, then a degree d equation defines a Riemann surface which is topologically a torus with g holes. There is a profound but easily described connection between the degree of a curve’s equation d and the genus g of its Riemann surface. This is given by the degreegenus formula which states that where g is the genus of the Riemann surface and d is the degree of the curve’s equation. Remembering the examples we have met, note that for gives , a sphere, and for gives , a torus. For curves with singular points, the formula can be generalized including a correction term for each singularity, as shown by Max Noether in 1884. This is a first glimpse at some of the deep connections between topology, algebra, geometry, and calculus that would be uncovered in mathematics. Quoting the French mathematician Jean Dieudonné, ‘In the history of mathematics the twentieth century will remain as the century of topology.’ The century would see a fledgling subject, intuitively but informally understood, go on to become one of the central pillars of mathematics. It is worth noting that those early topologists—Möbius, Klein, Riemann—did not in their time have available the rigorous definitions necessary to prove their results to modern standards. In 1861 Möbius gave an early sketch proof of the classification theorem for orientable surfaces, and Walther Von Dyck gave a sketch proof for all closed surfaces in 1888. But without having any formal definition of what a surface is, these proofs can at best be considered incomplete. This is not to relegate such proofs to the dustbin, nor to consider them simply wrong, as such proofs often contain most or all of the crucial ideas of a proof. Somewhat differently expressed rigorous versions of the classification theorem would be proved by Max Dehn and Poul Heegaard in 1907 and by Roy Brahana in 1921. Curves and surfaces are one and twodimensional examples of manifolds, spaces that look ‘up close’ like the real line, the plane, or some higher dimensional equivalent. It wasn’t until 1936 that Hassler Whitney gave the modern definition of a manifold, and proved an important theorem showing when manifolds can be embedding in space (recall, for example, how the Klein bottle cannot be made in 3D space without selfintersections but can be made in 4D). An important aspect of modern geometry concerns the different types of mathematical structure—continuous, smooth, complex, metric—that can be put on these manifolds and I will say a little more on this in Chapter 5 when we discuss differential topology. Chapter 3 Thinking continuously Given just one sentence for the task, many topologists might choose to describe their subject as the study of continuity. The word ‘continuous’ appeared a few times in Chapter 1 but it was left to the reader’s intuition as to quite what the word entailed. In many ways this reflects how mathematicians used to regard continuity—historically it was just considered evident what was meant by ‘continuous’ and as many (but not all) of the results about continuous functions that are ‘obvious’ also happen to be true, relatively little effort was spent providing further clarity. A rigorous definition of continuity did not appear until the 19th century. In your everyday routine there are continuous and discontinuous functions around you. For example, if you drive to work, the distance you have travelled after a certain time will be a continuous function of time—for this not to be the case would mean that at one moment your car was in a certain place only for it to immediately afterwards be at another place some distance away. Your speed on the journey will similarly be continuous. However, the acceleration need not be; if you were sat at rest (say at traffic lights) the acceleration would be zero but then would jump to a certain value once your foot was on the accelerator. The graphs in Figure 22 give a plausible (if simplistic) model for someone’s drive to work. 22. Distance, speed, and acceleration on a journey (a) Distance, (b) Speed, (c) Acceleration. From Figure 22(b) we can see that the car stops at t3—where the speed s(t) becomes zero—and after t4 increases to the speed limit. The distance travelled d(t) in Figure 22(a) is a continuous function of time t. Historically this would have been understood as meaning its graph could be drawn without taking pen from paper, but we will seek to provide a fuller understanding. But the acceleration function a(t) is not continuous because of the jumps in the graph in Figure 22(c). The times t1, t2, … t6 of discontinuity in the acceleration relate to the driver’s foot coming off the accelerator, being put on the brake, coming off the brake, and then the pattern repeats again. My aim in this chapter is to provide a more rigorous sense of just what continuity entails for realvalued functions of a real variable. This means we will focus on functions having a single numerical input and a single numerical output. Functions The idea of a function is a central one to mathematics, though this has only been true since around the 17th century. Once Descartes and Fermat independently introduced the idea of Cartesian coordinates x and y to describe position in a plane, a curve could just as easily be described by an equation as by its geometry. For example, the curve is a parabola, a curve the ancient Greeks would have investigated solely using geometry. A sketch of this curve is given in Figure 23(a)—the curve’s equation gives a rule for plotting, above each point (x, 0) of the xaxis, a point (x, x2). Note how certain algebraic properties of the function are represented in the shape and position of the curve—as x2 ⩾ 0 for all x, the curve lies entirely on or above the xaxis; as , the curve is symmetric about the yaxis. 23. Examples of graphs (a) Graph of , (b) Graph of y =. For some functions, we might naturally have to limit the allowed inputs—or we might choose to do so anyway. For example, if then we at least need to ensure that x is nonzero as division by zero is meaningless; for the function , then we cannot permit x to be negative, as no real number has a negative square (Figure 23(b)). More generally, a function comes with a set of inputs, known as the domain, and there is likewise the codomain, a set containing the outputs. It is an important, if subtle, point to appreciate that a function is this whole package: the domain, the codomain, and the rule assigning values. Some first thoughts about continuity Let’s first try to understand what it means for a function, with real inputs and outputs, to be continuous. Currently we sort of intuitively know continuity when we see it. Certainly, looking at two functions in Figures 24(a) and 24(b), it seems reasonable to say f(x) is continuous and g(x) is not continuous, and further that g(x) is discontinuous only at . (The full disc on the graph shows where the function takes its value, so that .) But what does intuition say about Figure 24(c)? Is h(x) continuous or not? It seems that, if h(x) is discontinuous, the only point of discontinuity is , but the function oscillates so wildly there, we may now be thinking that our intuition didn’t have all the answers. 24. Continuous and discontinuous functions (a) y = f (x) = sin x, (b) , (c) . Back to Figure 24(b), what is it about the function’s behaviour at that makes us think g(x) is discontinuous? For input x a little more than 1, then g(x) has much the same value as g(1); however for input x a little less than 1 then g(x) is noticeably different from g(1). It is this jump in output at 1 that is crucial to g(x) being discontinuous at . At first, we might be tempted to think this is because g(1) is different from the value of g(x) achieved immediately before we get to x equalling 1. But there are all sorts of problems with this thinking. First, there is no real number x that is ‘immediately before’ 1. Given a number like 0.999, close to 1, then we can always improve on that and see 0.9999 is a little closer. Or we might suggest using 0.999 … (where the ellipsis means that there are infinitely many recurring 9s) but this is just another decimal expansion for 1. More rigorously, for any input x < 1 then (1+x)/2 is less than 1 but closer to 1 than x is. Instead we might be tempted to talk about an input that is infinitesimally close to 1 but then—whatever we mean by this—we are no longer talking about the real numbers and have just replaced resolving one definition with resolving a different one. We need another approach that can be comfortably expressed entirely in terms of real numbers. This problem was independently resolved in the 19th century by Bernard Bolzano and Karl Weierstrass. We feel that g(x) is discontinuous at because g(x) is noticeably different from g(1) for some inputs x nearby to 1. There is still quite a bit of subtlety needed to fully capture what this means. In our example, g(x) has a jump of 1 from output values near 2 (just before ) to output values near 1 (just after ). The size of that jump was unimportant, the presence of any jump at all was sufficient. And the notion of ‘nearby inputs’ should not be interpreted as several inputs that are in some sense close to 1; rather we mean there are inputs x arbitrarily close to 1 such that g(x) is noticeably different from g(1). Necessarily this means that we are talking about infinitely many such inputs x, not just several x. By way of example, it is enough to note that: This rigorously shows that g(x) is discontinuous at . The sequence of inputs 0.9, 0.99, 0.999, 0.9999, … gets arbitrarily close to 1. What this means is: however demanding ‘nearby to 1’ is required to be, there are inputs from this sequence that are at least that close. Whilst we still haven’t quite defined just what we mean by discontinuous, we have made some progress with regard to the function h(x) (Figure 24(c)). This function does not appear to have any noticeable ‘jump’ in outputs, but it does seem to meet the definition h(x) is noticeably different from h(0) for some inputs x arbitrarily near to 0. Near the function h(x) is varying crazily. From the graph we can see that there are inputs x, arbitrarily close to 0, where whilst we have . It now seems clear by our emerging sense of continuity that h(x) is discontinuous at . An example in detail We still need to be careful turning these nascent thoughts into a rigorous definition. We’ll consider in detail the function which is continuous for all inputs x. If f(x) is continuous at an input then—based on our previous thoughts—we need that f(x) is not noticeably different from f(a) for all inputs x suitably near to a. Take a moment to appreciate why we need all inputs x suitably near to a to produce not noticeably different outputs f(x) to f(a). If some—but only one—nearby input x to a resulted in noticeably different outputs f(x) and f(a), then we could just tighten our notion of ‘suitably near’ to exclude the problem input x. In fact, if we can never get ‘suitably near’ with our inputs, then this means that there were arbitrarily close problematic inputs x to a where f(x) was noticeably different from f(a)—so, f(x) would be discontinuous at . To begin, what does it mean for to be continuous at ? Is it true that x2 is not noticeably different from for all inputs x suitably near to 0? We try out some values in Table 3. Table 3. Sample input and output values for It seems—admittedly only on the basis of five choices of x—that x2 is closer to 0 than x is to 0, and some quick algebra checks that small numbers generally square to smaller numbers (in magnitude). We cannot find inputs x close to 0 where the outputs x2 and 0 are noticeably different. Now we can hang some rigorous mathematics on these initial thoughts: whatever potential ‘noticeable difference’ in the outputs x2 and we consider, represented by a positive number e, then there need to be ‘suitably close’ inputs x to 0, represented by a positive number d, such that if inputs x and 0 differ by less than d then outputs x2 and 0 differ by less than e. As the outputs here are closer to one another than the inputs are—that is, as x2 is closer to 0 than x is to 0—then we can just choose d to equal e. So if inputs differ by e or less so do the outputs. We have then shown that is continuous at . What about continuity at a different input, say ? We can create a similar table to Table 3 (see Table 4). Table 4. More sample input and output values for The function is growing much more rapidly at than it is at . A change of around 0.1 in the inputs leads to a change in the outputs of around 200; a change of 0.01 in the inputs still leads to a difference of around 20 in the outputs. This may lead you to think that the outputs are ‘noticeably different’ here, but a more careful check of other inputs would show that these large differences have been incrementally achieved. All this is a consequence of the function changing more rapidly near , and what needs tightening is our notion of ‘suitably near’. As the function is growing more rapidly, small changes in the input will lead to relatively large changes, but still in a continuous fashion. If we consider the input , a little larger than the input 1000, then the difference in the outputs equals as d2 < d when d < 1. So a shift in inputs by d results in a shift of outputs roughly 2000 times larger. (Note similar behaviour in Table 4.) This is, in itself, not a problem but it does mean that if we want the outputs to differ by no more than e then we should only allow the inputs to differ by no more that e/2001. This still shows the continuity of at , we just needed a tighter sense of ‘suitably near’ with the inputs as the function was growing so fast. For continuity at yet larger inputs that notion would have to become yet more stringent, but we would always be able to find some small wiggle room about an input for which the outputs don’t differ beyond the desired amount e. A rigorous definition Putting all this thinking together gives us a rigorous definition of continuity. I’d suggest reading the definition and seeking to understand how this means that the function in Figure 25(a) is continuous and the one in Figure 25(b) isn’t, but if you find the generality of the definition and the technical level of the language difficult then move on to the next section on the properties of continuous functions. And be reassured, as it took generations of mathematicians to finally get this definition right, and current and past generations of mathematics undergraduates still wrestle with proofs involving this definition in their analysis courses. 25. The rigorous definition of continuous and discontinuous (a) A continuous function, (b) A discontinuous function. Formally, then, a function with real inputs x and real outputs f(x) is continuous at an input if: for any positive e there is some positive d such that the difference between the outputs f(x) and f(a) is less than e when the difference between the inputs x and a is less than d. In Figure 25(a), we are focusing on demonstrating the continuity of f(x) at input . A particular choice of e > 0 has been made and our task now is to make sure that the outputs don’t differ from f(a) by more than this e. So the outputs have to remain below f(a) + e and above f(a) – e (as shown on the yaxis). And this has to happen for inputs x in some range a – d < x < a+d. We can see from Figure 25(a) that some such interval has been found, as shown on the xaxis—the range of outputs on this interval are bounded by the dashed lines and these fall within t