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Jiˇr´ı Matouˇsek Using the Borsuk–Ulam Theorem Lectures on Topological Methods in Combinatorics and Geometry Writtenincooperationwith AndersBjo¨rnerandGu¨nterM.Ziegler 2nd,correctedprinting Jiˇr´ıMatouˇsek CharlesUniversity DepartmentofAppliedMathematics Malostranske´na´m.25 11800Praha1 CzechRepublic [email protected] Corrected2ndprinting2008 ISBN978-3-540-00362-5 e-ISBN978-3-540-76649-0 Universitext LibraryofCongressControlNumber:2007937406 MathematicsSubjectClassification(2000):05-01,52-01,55M20;05C15,05C10,52A35 (cid:1)c 2003Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstoragein databanks.Duplicationofthispublicationorpartsthereofispermittedonlyunderthepro- visionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andper- missionforusemustalwaysbeobtainedfromSpringer.Violationsareliabletoprosecution undertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com Preface A number of important results in combinatorics, discrete geometry, and the- oretical computer science have been proved by surprising applications of al- gebraic topology. Lova´sz’s striking proof of Kneser’s conjecture from 1978 is amongthefirstandmostprominentexamples,dealingwithaproblemabout finite sets with no apparent relation to topology. During the last two decades, topological methods in combinatorics have become more elaborate. On the one hand, advanced parts of algebraic topol- ogy have been successfully applied. On the other hand, many of the earlier results can now be proved using only fairly elementary topological notions andtools,andwhilethefirsttopologicalproofs,likethatofLova´sz,aremas- terpieces of imagination and involve clever problem-specific constructions, reasonably generalrecipesexistatpresent.Forsometypesof problems,they suggest how the desired result can be derived from the nonexistence of a certainmap(“testmap”)betweentwotopologicalspaces(the“configuration space” and the “target space”). Several standard approaches then become availableforprovingthenonexistenceofsuchamap.Still,thenumberofdif- ferentcombinatorialresultsestablishedtopologicallyremainsrelativelysmall. This book aims at making elementary topological methods more easily accessible to nonspecialists in topology. It covers a number of substantial combinatorial and geometric results, and at the same time, it introduces the required material from algebraic topology. Background in undergraduate mathematics is assumed, as well as a certain mathematical maturity, but no priorknowledgeofalgebraictopology.(Butlearningmorealgebraictopology fromothersourcesiscertainlyencouraged;thistextisnosubstituteforproper foundations in that subject.) We concentrate on topological tools of one type, namely, the Borsuk– Ulam theorem and similar results. We develop a systematic theory as far as our restricted topological means suffice. Other directions of research in topological methods, often very beautiful and exciting ones, are surveyed in Bj¨orner [Bjo¨95]. Historyandnotesonteaching. ThistextstartedwithacourseItaughtin fall1993inPrague(amotivationforthatcourseismentionedinSection6.8). Transcripts of the lectures made by the participants served as a basis of the first version. Some years later, a course partially based on that text was vi Preface taughtbyGu¨nterM.Ziegler inBerlin.Hemadeanumberofcorrectionsand additions(inthepresentversion,thetreatmentofBierspheresinSection5.6 is based on his writing, and Chapters 1, 2, and 4 bear extensive marks of his improvements).Thepresentbookisessentiallyathoroughlyrewrittenversion preparedduringapredoctoralcourseItaughtinZu¨richinfall2001,withafew thingsaddedlater.Mostofthematerialwascoveredinthecourse:Chapter1 was assigned as introductory reading, and the other chapters were presented in approximately 25 hours of teaching, with some omissions throughout and only a sketchy presentation of the last chapter. The material of this book should ultimately become a part of a more extensive project, a textbook of “topological combinatorics” with Anders Bj¨orner (the spiritual father of the project) and Gu¨nter M. Ziegler as coau- thors. A substantial amount of additional text already exists, but it appears that finishing the whole project might still take some time. We thus chose to publish the present limited version, based on my lecture notes and revolving around the Borsuk–Ulam theorem, separately. Although Anders and Gu¨nter decided not to be “official” coauthors of this version, the text has certainly benefited immensely from discussions with them and from their insightful comments. Sources. The 1994 version of this text was based on research papers, on a thorough surveyof topological methods in combinatorics byBjo¨rner[Bjo¨95], and on a survey of combinatorial applications of the Borsuk–Ulam theorem by Ba´ra´ny [Ba´r93]. The presentation in the current version owes much to the recent handbook chapter by Zˇivaljevi´c [Zˇiv04] (an extended version of [Zˇiv04] is [Zˇiv96]). The continuation [Zˇiv98] of that chapter deals with more advanced methods beyond the scope of this book. For learning algebraic topology, many textbooks are available (although in this subject it is probably much better to attend good courses). The first steps can be made with Munkres [Mun00] (which includes preparation in generaltopology)orStillwell[Sti93].Averygoodandreliablebasictextbook is Munkres [Mun84], and Hatcher [Hat01] is a vividly written modern book reaching quite advanced material in some directions. Exercises. This book is accompanied by 114 exercises; many of them serve as highly compressed outlines of interesting results. Only some have actually been tried in class. Theexerciseswithoutastarhaveshortsolutions,andtheyshouldusually be doable by good students who understand the text, although they are not necessarilyeasy.Allotherexercisesaremarkedwithastar:themorelaborious onesand/orthoserequiringanonobviousidea.Eventhisroughclassification is quite subjective and should not be taken very seriously. Acknowledgments. BesidesthealreadymentionedcontributionsofGu¨nter M. Ziegler and Anders Bjo¨rner, this book benefited greatly from the help of other people. For patient answers to my numerous questions I am much Preface vii indebted to Rade Zˇivaljevi´c and Imre B´ara´ny. Special thanks go to Yuri Ra- binovichforaparticularlycarefulreadingandalargenumberofinspiringre- marksandwell-deservedcriticisms.IwouldliketothankImreBa´ra´ny,P´eter Csorba, Allen Hatcher, Toma´ˇs Kaiser, Roy Meshulam, Karanbir Sarkaria, and Torsten Scho¨nborn for reading preliminary versions and for very useful comments. The participants of the courses (in Prague and in Zu¨rich) pro- videdastimulatingteachingenvironment,aswellasmanyvaluableremarks. I also wish to thank everyone who participated in creating the friendly and supportiveenvironmentsinwhichIhavebeenworkingonthebook.Theend- of-proof symbol is based on a photo of the European badger (“borsuk” in Polish) by Steve Jackson, and it used with his kind permission. Errors. If you find errors in the book, especially serious ones, I would appreciateitifyouwouldletmeknow(email:[email protected]). I plan to post a list of errors at http://kam.mff.cuni.cz/~matousek. Prague, November 2002 Jiˇr´ı Matouˇsek On the second printing. This is a revised second printing of the book. Errors discovered in the first printing have been removed, few arguments have been clarified and streamlined, and some new pieces of information on developments in the period 2003–2007 have been inserted. Most notably, a brief treatment of the cohomological index and of the Hom complexes of graphs is now included. For valuable comments and suggestions I’d like to thank Jos´e Rau´l Gonza´lesAlonso,BenBraun,P´eterCsorba,EhudFriedgut,DmitryFeichtner- Kozlov, Nati Linial, Mark de Longueville, Haran Pilpel, Mike Saks, Lars Schewe, Carsten Schultz, Ga´bor Simonyi, Ga´bor Tardos, Robert Vollmert, Uli Wagner, and Gu¨nter M. Ziegler. Prague, August 2007 J.M. Contents Preface ....................................................... v Preliminaries ................................................. xi 1. Simplicial Complexes ..................................... 1 1.1 Topological Spaces...................................... 1 1.2 Homotopy Equivalence and Homotopy .................... 4 1.3 Geometric Simplicial Complexes.......................... 7 1.4 Triangulations ......................................... 10 1.5 Abstract Simplicial Complexes ........................... 13 1.6 Dimension of Geometric Realizations...................... 16 1.7 Simplicial Complexes and Posets ......................... 17 2. The Borsuk–Ulam Theorem .............................. 21 2.1 The Borsuk–Ulam Theorem in Various Guises.............. 22 2.2 A Geometric Proof ..................................... 30 2.3 A Discrete Version: Tucker’s Lemma ...................... 35 2.4 Another Proof of Tucker’s Lemma ........................ 42 3. Direct Applications of Borsuk–Ulam...................... 47 3.1 The Ham Sandwich Theorem ............................ 47 3.2 On Multicolored Partitions and Necklaces ................. 53 3.3 Kneser’s Conjecture .................................... 57 3.4 More General Kneser Graphs: Dol’nikov’s Theorem ......... 61 3.5 Gale’s Lemma and Schrijver’s Theorem ................... 64 4. A Topological Interlude................................... 69 4.1 Quotient Spaces........................................ 69 4.2 Joins (and Products).................................... 73 4.3 k-Connectedness ....................................... 78 4.4 Recipes for Showing k-Connectedness ..................... 80 4.5 Cell Complexes ........................................ 82 x Contents 5. Z2-Maps and Nonembeddability .......................... 87 5.1 Nonembeddability Theorems: An Introduction ............. 88 5.2 Z2-Spaces and Z2-Maps ................................. 92 5.3 The Z2-Index ......................................... 95 5.4 Deleted Products Good ... .............................. 108 5.5 ...Deleted Joins Better ................................. 112 5.6 Bier Spheres and the Van Kampen–Flores Theorem......... 116 5.7 Sarkaria’s Inequality .................................... 121 5.8 Nonembeddability and Kneser Colorings................... 124 5.9 A General Lower Bound for the Chromatic Number......... 128 6. Multiple Points of Coincidence ........................... 145 6.1 G-Spaces .............................................. 145 6.2 E G Spaces and the G-Index ............................ 149 n 6.3 Deleted Joins and Deleted Products ...................... 157 6.4 The Topological Tverberg Theorem....................... 161 6.5 Many Tverberg Partitions ............................... 165 6.6 Necklace for Many Thieves .............................. 167 6.7 Z -Index, Kneser Colorings, and p-Fold Points ............. 170 p 6.8 The Colored Tverberg Theorem .......................... 174 A Quick Summary............................................ 179 Hints to Selected Exercises ................................... 185 References.................................................... 187 Index......................................................... 203 Preliminaries Thissectionsummarizesratherstandardmathematicalnotionsandnotation, and it serves mainly for reference. More special notions are introduced grad- ually later on. Sets. If S is a set, |S| denotes the number of elements ((cid:1)ca(cid:2)rdinality) of S. By2S wedenotethesetofallsubsetsof S (thepowerset); S standsforthe (cid:1) (cid:2) (cid:3)k (cid:1) (cid:2) set of all subsets of S of cardinality exactly k; and S = k S . We use ≤k i=0 i [n] to denote the finite set {1,2,...,n}. The letters R, C, Q, and Z stand for the real numbers, the complex numbers, the rational numbers, and the integers, respectively. By id we denote the identity mapping on a set X, with id (x)=x for X X all x∈X. Geometry. The symbol Rd denotes the Euclidean space of dimension d. PointsinRd aretypesetinboldface,andtheyareunderstoodasrowvectors; thus, we write x=(x1,...,xd)∈Rd. We write e1,e2,...,ed for the vectors of the standard orthonormal basis of Rd (e has a 1 at position i and 0’s i elsewhere). The scalar product of two vectors x,y ∈ Rd is (cid:2)x,y(cid:3)(cid:4)= xyT = x(cid:4)1y1 +x2y2 +···+xdyd. The Euclidean norm of x is (cid:4)x(cid:4) = (cid:2)(cid:1)x,x(cid:3) = x21+···+x2d.O(cid:2)ccasionallywealsoencounterthe(cid:1)p-norm(cid:4)x(cid:4)p = |x1|p+ |x2|p +···+|xd|p 1/p, 1 ≤ p < ∞, and the (cid:1)∞-norm (or maximum norm) (cid:4)x(cid:4)∞ =max{|x1|,|x2|,...,|xd|}. A hyperplane in Rd is a (d−1)-dimensional affine subspace, i.e., a set of the form {x ∈ Rd : (cid:2)a,x(cid:3) = b} for some nonzero a ∈ Rd and some b ∈ R. A (closed) half-space has the form {x ∈ Rd : (cid:2)a,x(cid:3) ≤ b}, with a and b as before. The unit ball {x ∈ Rd : (cid:4)x(cid:4) ≤ 1} is denoted by Bd, while Sd−1 = {x ∈ Rd : (cid:4)x(cid:4) = 1} is the (d−1)-dimensional unit sphere (note that S2 lives in R3!). AsetC ⊆Rd isconvexifforeveryx,y ∈C,thesegmentxy iscontained in C. The convex hull of a set X ⊆ Rd is the intersection of all convex sets containing X, and it is denoted by conv(X). Each point x ∈ conv(X) can be written as a convex combination of points of X: Ther(cid:5)e are points x1,x2,..(cid:5).,xn ∈ X and real numbers α1,...,αn ≥ 0 such that ni=1αi = 1 and x= n α x (if X ⊆Rd, we can always choose n≤d+1). i=1 i i

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