Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry

221 Pages·2003·1.72 MB·English

by  Jiri Matousek, A. Björner, G.M. Ziegler

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
[e-mail]
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
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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:[e-mail]).
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