Algorithmic Topology and Classification of 3-Manifolds

2007·9.08 MB·English

by  Sergei Matveev

Algorithms and Computation
in Mathematics Volume 9
•
Editors
ArjehM.Cohen HenriCohen
DavidEisenbud MichaelF.Singer BerndSturmfels
Sergei Matveev
Algorithmic Topology
and Classification
of 3-Manifolds
y
Second Edition
With264Figuresand36Tables
ABC
Author
Sergei Matveev
Chelyabinsk State University
Kashirin Brothers Street, 129
Chelyabinsk 454021
Russia
E-mail:[email protected]
LibraryofCongressControlNumber:2007927936
MathematicsSubjectClassification(2000):57M
ISSN1431-1550
ISBN 978-3-540-45898-2SpringerBerlinHeidelbergNewYork
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Preface to the First Edition
The book is devoted to algorithmic low-dimensional topology. This branch of
mathematics has recently been undergoing an intense development. On the
one hand, the exponential advancement of computer technologies has made
it possible to conduct sophisticated computer experiments and to implement
algorithmic solutions, which have in turn provided a motivation to search for
new and better algorithms. On the other hand, low-dimensional topology has
receivedanadditionalboostbecauseofthediscoveryofnumerousconnections
with theoretical physics.
There is also another deep reason why algorithmic topology has received
alotofattention.Itisthatasearchforalgorithmicsolutionsgenerallyproves
tobearichsourceofwell-stated mathematical problems.Speaking outofmy
experience, it seems that an orientation towards “how to” rather than just
“how is” serves as a probing stone for choosing among possible directions of
research – much like problems in mechanics led once to the development of
calculus.
It seemed to me, when planning this book, that I had an opportunity to
offeracoherentandreasonablycompleteaccountofthesubjectthatneverthe-
less would be mainly accessible to graduate students. Almost all parts of the
book are based upon courses that I gave at different universities. I hope that
thebookinheritsthestyleofalivelecture.Elementaryknowledgeoftopology
andalgebraisrequired,butunderstandingtheconceptsof“topologicalspace”
and “group” is quite sufficient for most of the book. On the other hand, the
bookcontainsalotofnewresultsandcoversmaterialnotfoundelsewhere,in
particular, the first proof of the algorithmic classification theorem of Haken
manifolds. It should be therefore useful to all mathematicians interested in
low-dimensional topology as well as to specialists.
Most of the time, I consider 3-manifolds without geometric structures.
There are two reasons for that. On the one hand, I prefer to keep the
exposition within the limits of elementary combinatorial approach, which is a
naturalenvironmentforconsideringalgorithmicquestions.Ontheotherhand,
geometric approach, which become incorporated into 3-manifold topology
VI Preface to the First Edition
after works of Thurston, is presented in mathematical literature comprehen-
sively [62,100,120,121]. Also, I touch upon computer investigation of hyper-
bolic manifolds only briefly, since this subject is completely covered by the
outstanding program SNAPPEA of Weeks [44].
When I embarked on this project I had not intended the writing of this
book to take so long. But once the ground rules were set up, I had no choice.
The rules are:
1. The book should be maximally self-contained. Proofs must be complete
and divided into “easy-to-swallow” pieces. Results borrowed from other
sourcesmustbeformulatedexactlyinthesameformastheyappearthere,
and no vague references to technique or proofs are allowed. The sources
must be published in books or journals with world-wide circulation.
2. Thetextshouldbewrittensothatateachmomentthereadercouldknow
what is going on. In particular, statements must always precede proofs,
not vice versa. Also, proofs must be straightforward, whenever possible.
3. Different parts of the book (sections, subsections, and even individual
statements) must be as independent as possible, so that one could start
reading the book from any point (after taking a few steps backwards or
looking at the index, if necessary, but without having to read everything
else before).
4. Each mathematical text is a cipher to encode information. The main dif-
ficulty for the reader often lies not in understanding the essence of a
statement, but in decoding what the authors really want to say. There-
fore, each paragraph should contain a redundant amount of information
to avoid misunderstanding. In particular, rephrasing is welcome.
Consistently with the algorithmic viewpoint, the book begins with a
comprehensive overview of the theory of special spines. The latter encode
3-manifolds in a rather comfortable, user-friendly, way, which also is easily
turned into a computer presentation. This chapter also contains an existence
result giving a criterion for two special polyhedra to be spines of the same
manifold.
Whilespecialspinesallowustoworkwithasinglemanifold,thetheoryof
complexity(Chap.2)attemptstooverviewthewholesetof3-manifoldsintro-
ducinganorderintotheirchaos.Specifically,thesetof3-manifoldsissupplied
with a filtration by finite subsets (of 3-manifolds of a bounded complexity),
and this allows us to break up the classification problem for all 3-manifolds
intoaninfinitenumberofclassificationproblemsforsomefinitesubsets.This
approach is implemented in Chap.7, where we describe a way to enumer-
ate manifolds of a given complexity. To be precise, we describe a computer
program that enumerates manifolds and conducts a partial recognition. The
finalrecognitionisdonebycomputingfirsthomologygroupsandTuraev–Viro
invariants. The latter are described in Chap.8; the exposition was intention-
allymadeaselementaryandprerequisite-freeaspossible.Theresultingtables
Preface to the First Edition VII
ofmanifoldsuptocomplexity6,oftheirminimalspecialspines,andofvalues
of their Turaev–Viro invariants are given in Appendix.
Chapter 3 contains the first ever complete exposition of Haken’s normal
surfacetheory,whichisthecornerstoneofalgorithmic3-dimensionaltopology.
Almost all known nontrivial algorithms in low-dimensional topology use it or
at least are derived from it. Several key algorithms are included into Chap.4.
In Chap.5 the Rubinstein–Thompson algorithm for recognition of S3 is pre-
sented.MyapproachisinasensedualtotheoriginalproofofThompsonand
seems to me more transparent.
Chapter 6 is the central part of the book. There, I prove the algorithmic
classification theorem for Haken manifolds. Surprisingly, although already in
1976 it was broadly announced that the theorem is true [58,59,131], no proof
appeared until 1995. Moreover, it turned out that the ideas described in the
above-mentioned survey papers, in Hemion’s book [42], and in other sources
are insufficient. I prove the theorem using some facts from the algorithmic
version of the Thurston theory of surface homeomorphisms [9].
For closed Haken manifolds other proofs of the algorithmic classification
theorem can be composed now. They are based on Thurston’s hyperboliza-
tion theorem for Haken manifolds containing no essential tori and annuli and
on Sela results on algorithmic recognition of hyperbolic manifolds [114,115].
A brief survey is contained in [73], where the author explains how a hyper-
bolic structure on a 3-manifold can be constructed algorithmically once one
is known to exist.
This book began with several lectures that I gave first at Tel-Aviv Uni-
versity in October–December 1990 and then at the Hebrew University of
Jerusalem in October 1991–January 1992. The lectures were extended to a
lecture course that I read in 1993 at the University of British Columbia as
a part of the Noted Scholar Summer School Program. Chapters 1, 2, and 8
are based on the notes taken by Djun Kim and Mark MacLean; my thanks
to them. Later on I returned regularly to these notes and extended them
by including new parts, once I had found what I hoped was the right way
to expose them. Lectures on the subject which I gave at Pisa University in
1998, 2002 and in the J.-W. Goethe Univesita¨t Frankfurt in April–May 2000
were especially valuable for me; you always profit when an attentive audience
wishestounderstandalldetailsandforcesthelecturertofindthemostprecise
arguments. I hope I have managed to capture the spirit of those lectures in
this text.
Many mathematicians have helped me during my work on the book.
Among them are M. Boileau, A. Cavicchioli, N. A’Campo, I. Dynnikov,
M. Farber, C. Hog-Angeloni, A. Kozlowski, W. Metzler, M. Ovchinnikov,
E. Fominykh, E. Pervova, C. Petronio, M. Polyak, D. Rolfsen,
A. Shumakovich, M. Sokolov, A. Sossinsky, H. Zieschang.
There are two more persons I wish to mention separately, my teacher
A. Chernavskii, who introduced me to the low-dimensional topology, and
VIII Preface to the First Edition
my colleague A. Fomenko, whose outstanding personality and mathematical
books influenced significantly the style of my thinking and writing.
A great part of this book was written during my stay at Max-Planck-
Institut fu¨r Mathematik in Bonn. I am grateful to the administration of the
institute for hospitality and for a friendly and creative atmosphere. I also
thanktheRussianFundofBasicResearchandINTASforthefinancialsupport
of my research.
Of course, the book could not even have been started without the encour-
aging support of Chelyabinsk State University, which is my home university.
I am profoundly grateful to all my colleagues for their help. Special thanks to
my beloved wife L. Matveeva, who is also a mathematician, for her patience
and help.
Chelyabinsk, Sergei Matveev
March 2003
Preface to the Second Edition
The book has been revised, and some improvements and additions have been
made. In particular, in Chap.7 several new sections concerning applications
of the computer program “3-Manifold Recognizer” have been included.
March 2007 Sergei Matveev
Contents
1 Simple and Special Polyhedra ............................. 1
1.1 Spines of 3-Manifolds .................................... 1
1.1.1 Collapsing........................................ 1
1.1.2 Spines ........................................... 2
1.1.3 Simple and Special Polyhedra....................... 4
1.1.4 Special Spines .................................... 5
1.1.5 Special Polyhedra and Singular Triangulations ........ 10
1.2 Elementary Moves on Special Spines ....................... 13
1.2.1 Moves on Simple Polyhedra......................... 14
1.2.2 2-Cell Replacement Lemma......................... 19
1.2.3 Bubble Move ..................................... 22
1.2.4 Marked Polyhedra................................. 25
1.3 Special Polyhedra Which are not Spines .................... 30
1.3.1 Various Notions of Equivalence for Polyhedra ......... 31
1.3.2 Moves on Abstract Simple Polyhedra ................ 35
1.3.3 How to Hit the Target Without Inverse U-Turns ...... 43
1.3.4 Zeeman’s Collapsing Conjecture..................... 46
2 Complexity Theory of 3-Manifolds......................... 59
2.1 What is the Complexity of a 3-Manifold?................... 60
2.1.1 Almost Simple Polyhedra........................... 60
2.1.2 Definition and Estimation of the Complexity.......... 62
2.2 Properties of Complexity ................................. 67
2.2.1 Converting Almost Simple Spines into Special Ones.... 67
2.2.2 The Finiteness Property............................ 70
2.2.3 The Additivity Property ........................... 71
2.3 Closed Manifolds of Small Complexity ..................... 72
2.3.1 Enumeration Procedure ............................ 72
2.3.2 Simplification Moves............................... 74
2.3.3 Manifolds of Complexity ≤6........................ 76
XII Contents
2.4 Graph Manifolds of Waldhausen........................... 83
2.4.1 Properties of Graph Manifolds ...................... 83
2.4.2 Manifolds of Complexity ≤8 ........................ 89
2.5 Hyperbolic Manifolds .................................... 97
2.5.1 Hyperbolic Manifolds of Complexity 9 ............... 97
2.6 Lower Bounds of the Complexity ..........................100
2.6.1 Logarithmic Estimates ............................101
2.6.2 Complexity of Hyperbolic 3-Manifolds ...............104
2.6.3 Manifolds Having Special Spines with One 2-Cell ......105
3 Haken Theory of Normal Surfaces .........................107
3.1 Basic Notions and Haken’s Scheme ........................107
3.2 Theory of Normal Curves.................................110
3.2.1 Normal Curves and Normal Equations ...............110
3.2.2 Fundamental Solutions and Fundamental Curves ......114
3.2.3 Geometric Summation .............................115
3.2.4 An Alternative Approach to the Theory of Normal
Curves ...........................................119
3.3 Normal Surfaces in 3-Manifolds ...........................123
3.3.1 Incompressible Surfaces ............................123
3.3.2 Normal Surfaces in 3-Manifolds with Boundary Pattern 126
3.3.3 Normalization Procedure ...........................127
3.3.4 Fundamental Surfaces..............................134
3.3.5 Geometric Summation .............................135
3.4 Normal Surfaces in Handle Decompositions ................138
4 Applications of the Theory of Normal Surfaces ............147
4.1 Examples of Algorithms Based on Haken’s Theory...........147
4.1.1 Recognition of Splittable Links......................148
4.1.2 Getting Rid of Clean Disc Patches...................150
4.1.3 Recognizing the Unknot and Calculating the Genus
of a Circle in the Boundary of a 3-Manifold...........157
4.1.4 Is M3 Irreducible and Boundary Irreducible? .........160
4.1.5 Is a Proper Surface Incompressible and Boundary
Incompressible? ...................................163
4.1.6 Is M3 Sufficiently Large?...........................166
4.2 Cutting 3-Manifolds along Surfaces ........................176
4.2.1 Normal Surfaces and Spines ........................176
4.2.2 Triangulations vs. Handle Decompositions ............188
5 Algorithmic Recognition of S3 .............................191
5.1 Links in a 3-Ball ........................................192
5.1.1 Compressing Discs and One-legged Crowns ...........192
5.1.2 Thin Position of Links .............................195