Table Of ContentPreface
Attheturn ofthetwentiethcenturymathematicianshad amonopolyon well
defined complicated global problems, e.g. celestial mechanics, fixed points of
high dimensional nonlinear functions, the geometry of level sets of differen-
tiablefunctions,algebraicvarieties,distinguishingtopologicalspaces,etc.Al-
gebraic topology, of which homology is a fundamental part, was developed in
response to such challenges and represents one of the great achievements of
twentieth century mathematics. While its roots can be traced to the middle
of the nineteenth century with Euler’s famous formula that for the surface of
a convex polyhedron
faces − edges + vertices =2,
it is fair to say that it was began as a subject in its own right in the seminal
works of Henri Poincar´e on “Analysis Situs.” Though motivated by analytic
problems as his techniques developed they took on a combinatorial form sim-
ilar in spirit to Euler’s formula.
The power of algebraic topology lies in it coarseness. To understand this
statement consider Euler’s formula. Observe for instance that the size of the
polyhedron is of no importance. In particular, therefore, small changes in the
shape of the polyhedron do not alter the formula. On the other hand if one
beginswithapolyhedronthatispuncturedbyk holesasopposedtoaconvex
polyhedron, then the formula becomes
faces − edges + vertices =2−2k.
As a result, counting local objects, the faces, edges and vertices, of a polyhe-
dron allows us to determine a global property, how many holes it has. Fur-
thermore, it shows that a formulas of this form can be used to distinguish
objects with important different geometric properties.
The potential of Poincar´e’s revolutionary ideas for dealing with global
problems were quickly recognized and this led to a broad development of the
subject. However, as is to be expected, the form of development matched the
problemsofinterest.Asindicatedaboveatypicalquestionmightbeconcerned
VI Preface
with the structure of the level set of a differentiable function. Solving such
problemsusingpurelycombinatorialargumentssuggestedbyEuler’sformula,
i.e. cutting the set into a multitude of small pieces and then counting, is in
general impractical. This led to the very formidable and powerful algebraic
machinery that is now referred to as Algebraic Topology. In its simplest form
this tool takes objects defined in terms of traditional mathematical formu-
lae and produces algebraic invariants which provide fundamental information
about geometric properties of the objects.
As we begin the twenty-first century complexity has spread beyond the
realm of mathematics. With the advent of computers and sophisticated sens-
ing devices, scientists, engineers, doctors, social scientists, businessmen, all
have access to or through numerical simulation can create huge data files.
Furthermore, for some of these data sets the crucial information is geometric
innature,butitmakeslittlesensetothinkofthesegeometricobjectsasbeing
presented in terms of or derived from traditional mathematical formulae. As
an example think of medical imaging. Notice that even though the input is
different, the problems remain the same identifying and classifying geometric
propertiesorabnormalities.Furthermore,inherentinnumericalorexperimen-
tal data is error. Thus what is needed is a framework in which geometrical
objects can be recognized even in the presence of small perturbations.
Hopefully these arguments suggest that the extraordinary success of al-
gebraic topology in the traditional domains of mathematics can be carried
over to this new set of challenges. However, to do so requires the ability
to efficiently compute the algebraic topological quantities starting with ex-
perimental or numerical data - information which is purely combinatorial in
nature.
The purpose of this book is to present a computational approach to ho-
mology with the hope that such a theory will prove beneficial to the analysis
and understanding of today’s complex geometric challenges. Naturally this
meansthatourintendedaudienceincludescomputerscientists,engineers,ex-
perimentalists and theoreticians in nonlinear science. As such we have tried
tokeepthemathematicalprerequisitestoanabsoluteminimum.Atthesame
time we, the authors, are mathematicians and proud of our trade. We believe
that the most significant applications of the theory will be realized by those
who understand the fundamental concepts of the theory. Therefore, we have
insistedonarigorousdevelopmentofthesubject.Thus,thisbookcanalsobe
used as an introductory text in homology for mathematics students. It differs
fromthetraditionalintroductorytopologybooksinthatagreatdealofeffort
is spent on discussing the computational aspects of the subject.
The book is divided into three parts. Part I contains the core material.
It provides a rigorous introduction to the homology of spaces and continu-
ous functions and is meant to be read in sequence. Homology is a beautiful
subject in that topology is transformed into algebra and from the algebra
one can recover aspects of the topology. However, this process involves some
Preface VII
deep ideas and it is easy to lose track of the big picture in the midst of the
mathematical technicalities. With this in mind we have included two preview
chapters(Chapters1and5)wheretheideasaresketchedthroughverysimple
examples and without too much concern for rigor.
InChapter2wedefinehomologyandinvestigateitsmostelementaryprop-
erties. In particular, we explain how to each topological space we can assign
a sequence of abelian groups call the homology groups of the space. There
is a caveat and that is that the topological spaces we consider must be built
out of d-dimensional unit cubes with vertices on the integer lattice. This is in
contrast to the standard combinatorial approach which is based on simplices.
There are two reasons for this. The first comes from applications. Consider
a digital images. The basic building blocks are pixels which are easily iden-
tified with squares. Similarly an experimental or numerically generated data
point comes with errors. Thus an d-dimensional data point can be thought of
as lying inside a d-dimensional cube whose width is determined by the error
bounds. The second reason - which will be made clear in the text - comes
from the simplicity of certain algorithms.
InChapter3weshowthathomologyiscomputablebypresentingindetail
analgorithmbasedonlinearalgebraovertheintegers.Thisisessentialinthe
sense that it demonstrates that the homology group of a topological space
made up of cubes is computable. However, for spaces made up of many cubes
the algorithm is of little immediate practical value. Therefore, in Chapter 4
we introduce combinatorial techniques for reducing the number of elements
involved in the computation.
ThecontentsofChapter4alsonaturallyforeshadowquestionsconcerning
maps between topological spaces and maps between the associated homology
groups. The construction of homology maps is done in Chapter 6. Again, we
approach this problem from the point of view of multivalued maps. This has
an extremely important consequence. One can efficiently compute homology
maps-anabsolutelyessentialfeaturegiventhepurposesofthisendeavor.We
know of no practical algorithms for producing simplicial maps which approx-
imate continuous maps. Algorithms for the computation of these maps are
discussed in Chapter 7.
As was mentioned earlier we have attempted to keep the necessary pre-
requisites for this book to a minimum. Ideally the reader would have had an
introductory course in point-set topology, an abstract algebra course and fa-
miliarity with computer algorithms. On the other hand such a reader would
know far more topology, algebra and computer science than is really neces-
sary. Furthermore, the union of these subjects in the typical curriculum of
ourdesiredreadersisprobablyfairlyrare.Forthisreasonwehaveincludedin
Part III brief introductions to these subjects. Perhaps a more traditional or-
ganizationwouldhaveplaced Chapters12, 13, and 14 at the beginning of the
book.However,itisouropinionthatnothingkillstheexcitementofstudying
a new subject as rapidly as the idea of multiple chapters of preliminary ma-
VIII Preface
terial. We suggest that the reader begin with Chapter 1 and consult the last
three Chapters on a need to know basis.
We argued at the beginning of this preface that algebraic topology has an
important role to play in the analysis of numerical and experimental data.
In Part II we elaborate on these ideas. It should be mentioned that appli-
cations of homology to these types of problems is a fairly new idea and still
in a fairly primitive state. Thus the focus of this part is on conveying the
potential rather than elaborate applications. We begin in Chapter 8 with a
discussionthatrelatescubicalcomplexestoimagedataandnumericallygener-
ateddata.Eventhesimpleexamplespresentedheresuggesttheneedformore
sophisticated ideas from homology. Therefore, Chapter 9 introduces more so-
phisticated algebraic concepts and computational techniques that are at the
heart of homological algebra. In Chapter 10 we indicate how homology can
be used to study nonlinear dynamics. In some sense this is one of the most
traditional applications of the subject - the focus of much of Poincar´e’s work
was related to differential equations and, in fact, he set the foundations for
what is known today as dynamical systems. The modern twist is that these
ideas can be tied to numerical methods to obtain computer assisted proofs in
this subject. Finally, thought the cubical theory presented in this book has
concrete advantages it is also in many ways too rigid. Thus in Chapter 11 we
show that the cubical theory is equivalent to the simplicial theory, thereby
tying the results and techniques of this book to the enormous body of work
known as algebraic topology.
Preliminaryversionofthisbookhavealreadybeenusedbytheauthorsto
teach courses at a variety of levels: undergraduate and graduate; for a mixed
audienceofmathematics,computerscienceandengineeringstudents;asaone-
semestercourseorafullyearcourse.Aswementionedearlier,PartIismeant
as a core material. However, in a one-semester course, a teacher attempting
to follow closely the order of chapters could not arrive at applications or ex-
tensions presented in Part II. It is possible to jump directly to specific topics
in Part II without having completed Part I and the diagram in Figure 0.1
is indicating how. The arrows do not necessarily mean that the whole con-
tentsofarelatedchaptercanbetaughtafterskippingprecedingchapters.For
example, a teacher orienting his course towards applications may go immedi-
atelyfrom Chapter2toChapter8 orfromChapter6toChapter 10provided
that certain statements about properties and computation of homology are
temporarilytakenforgranted.Also,ateachermaycoverthefirsttwosections
of Chapter 11 concerning simplicial complexes right after Chapter 2 or even
parallel to it but the last section on the homology functor could only be at-
tained after having taught Chapter 6. Part III can be regarded either as an
appendixorasacorematerialdependingontheleveloftheaudienceandthe
profile of a course.
Many colleagues and students helped us by means of valuable comments,
detailed proofreading or by research contributions which affluence the con-
tents of the book. First, the research of our graduate students: shaped and
Preface IX
Part III Part I Part II
Chapter 1
(cid:63)
Chapter 12 (cid:45) Chapter 2 (cid:45) Chapter 8
(cid:64)
(cid:161)(cid:181)
(cid:161) (cid:64) (cid:162)(cid:184)
(cid:161) (cid:63) (cid:64) (cid:162)
(cid:161) (cid:64)(cid:162)
Chapter 13(cid:161) (cid:45) Chapter 3 (cid:162) (cid:64)
(cid:162) (cid:64)
(cid:162)(cid:184) (cid:162) (cid:64)
(cid:162) (cid:63) (cid:162) (cid:64)
(cid:162) (cid:162) (cid:64)
(cid:64)(cid:82)
(cid:162) Chapter 4 (cid:162) Chapter 11
(cid:162)
(cid:161)(cid:181)
(cid:162) (cid:161)
(cid:162) (cid:63) (cid:161)
(cid:162) (cid:161)
Chapter 14(cid:162) Chapter 5 (cid:161)
(cid:65) (cid:161)
(cid:65) (cid:161)
(cid:65) (cid:63) (cid:161) (cid:63)
(cid:65) (cid:161)
(cid:65) Chapter 6 (cid:161) (cid:45) Chapter 9
(cid:65) (cid:64)
(cid:65) (cid:64)
(cid:65) (cid:63) (cid:64) (cid:63)
(cid:65)(cid:85) (cid:64)
(cid:64)(cid:82)
Chapter 7 (cid:45) Chapter 10
Fig. 0.1. Chapter Dependence Chart
motivated the whole project of writing this book. Teaching this material was
exciting and challenging due to the freshness of results presented and the
interplay between the mathematical rigor, computations, and applications.
Thus the feedback we got from students enrolled in our courses played a very
important role in improving the text. We are particularly indebted to: for
a detailed proofreading of the text at its early stages.
X Preface
Atlanta, Krak´ow, Sherbrooke, Tomasz Kaczynski
January 2003 Konstantin Mischaikow
Marian Mrozek
Contents
Part I Homology
1 Preview of Homology...................................... 3
1.1 Graphs................................................. 4
1.2 Topological and Algebraic Boundaries...................... 7
1.3 Keeping Track of Directions .............................. 12
1.4 Mod 2 Homology of Graphs............................... 15
2 Cubical Homology ......................................... 27
2.1 Cubical Sets ............................................ 28
2.1.1 Elementary Cubes................................. 28
2.1.2 Cubical Sets ...................................... 31
2.1.3 Elementary Cells .................................. 33
2.2 The Algebra of Cubical Sets .............................. 36
2.2.1 Cubical Chains.................................... 36
2.2.2 Cubical Chains in a Cubical Set..................... 42
2.2.3 The Boundary Operator............................ 44
2.2.4 Homology of Cubical Sets .......................... 49
2.3 Connected Components and H (X)........................ 55
0
2.4 Elementary Collapses .................................... 59
2.5 Acyclic Cubical Spaces................................... 68
2.6 Homology of Abstract Chain Complexes.................... 74
2.7 Reduced Homology ...................................... 78
3 Computing Homology Groups ............................. 83
3.1 Matrix Algebra over Z ................................... 84
3.2 Row Echelon Form ...................................... 96
3.3 Smith Normal Form .....................................105
3.4 Structure of Abelian Groups ..............................114
3.5 Computing Homology Groups.............................120
3.6 Computing Homology of Cubical Sets ......................121
XII Contents
3.7 Preboundary of a Cycle - Algebraic Approach...............127
4 Chain Maps and Reduction Algorithms....................129
4.1 Chain Maps ............................................129
4.2 Chain Homotopy ........................................136
4.3 Internal Elementary Reductions ...........................142
4.3.1 Elementary Collapses Revisited .....................142
4.3.2 Generalization of Elementary Collapses ..............144
4.4 KMS Reduction Algorithm ...............................152
5 Preview of Maps ..........................................159
5.1 Rational Functions and Interval Arithmetic .................160
5.2 Maps on an Interval .....................................161
5.3 Constructing Chain Selectors .............................170
5.4 Maps of Γ1 .............................................174
6 Homology of Maps ........................................185
6.1 Representable Sets.......................................185
6.2 Cubical Multivalued Maps................................190
6.3 Chain Selectors .........................................194
6.4 Homology of Continuous Maps ............................199
6.4.1 Cubical Representations............................200
6.4.2 Rescaling.........................................205
6.5 Homotopy Invariance ....................................214
7 Computing Homology of Maps ............................219
7.1 Producing Multivalued Representation .....................221
7.2 Chain Selector Algorithm.................................224
7.3 Computing Homology of Maps ............................225
7.4 Computing Homology Maps - Reduction Approach ..........226
7.4.1 Reduction of a Chain Map..........................226
7.4.2 Computing Homology of a Chain map ...............227
7.5 Geometric Preboundary Algorithm ........................227
Part II Extensions
8 Prospects in Digital Image Processing .....................239
8.1 Digital Images and Cubical Complexes .....................240
8.2 Complicated Time Dependent Patterns.....................245
8.3 Size Function ...........................................248
Contents XIII
9 Homological Algebra ......................................257
9.1 Relative Homology ......................................257
9.1.1 Relative Homology Groups .........................258
9.1.2 Maps in Relative Homology.........................263
9.2 Exact Sequences.........................................266
9.3 The Connecting Homomorphism ..........................270
9.4 Mayer-Vietoris Sequence .................................277
9.5 Weak Boundaries........................................281
10 Nonlinear Dynamics .......................................285
10.1 Maps and Symbolic Dynamics.............................286
10.2 Differential Equations and Flows ..........................295
10.3 Waz˙ewski Principle ......................................297
10.4 Fixed Point Theorems ...................................301
10.4.1 Fixed Points in The Unit Ball.......................301
10.4.2 Lefschetz Fixed Point Theorem .....................303
10.5 Degree Theory ..........................................309
10.5.1 Degree on Spheres.................................310
10.5.2 Topological Degree ................................313
10.6 Complicated Dynamics...................................319
10.6.1 Index Pairs and Index Map.........................320
10.6.2 Topological Conjugacy .............................333
10.7 Computing Chaotic Dynamics.............................337
11 Homology of Topological Polyhedra........................339
11.1 Simplicial Homology .....................................339
11.2 Comparison of Cubical and Simplicial Homology ............346
11.3 Homology Functor.......................................349
11.3.1 Category of Cubical Sets ...........................349
11.3.2 Category of Topological Polyhedra...................350
11.3.3 Connected Simple Systems .........................350
11.3.4 Homology of Topological Polyhedra..................352
Part III Tools from Topology and Algebra
12 Topology ..................................................355
12.1 Norms and Metrics in Rd.................................355
12.2 Topology ...............................................360
12.3 Continuous Maps........................................364
12.4 Connectedness ..........................................369
12.5 Limits and Compactness .................................373
XIV Contents
13 Algebra....................................................377
13.1 Abelian Groups .........................................377
13.1.1 Algebraic Operations ..............................377
13.1.2 Groups...........................................378
13.1.3 Cyclic Groups and Torsion Subgroup ................380
13.1.4 Quotient Groups ..................................382
13.1.5 Direct Sums ......................................383
13.2 Fields and Vector Spaces .................................385
13.2.1 Fields............................................385
13.2.2 Vector Spaces.....................................387
13.2.3 Linear Combinations and Bases .....................388
13.3 Homomorphisms ........................................391
13.3.1 Homomorphisms of Groups .........................391
13.3.2 Linear Maps ......................................394
13.3.3 Matrix Algebra ...................................395
13.4 Free Abelian Groups.....................................398
13.4.1 Bases in Groups...................................398
13.4.2 Subgroups of Free Groups ..........................403
13.4.3 Homomorphisms of Free Groups.....................405
14 Algorithm Presentation....................................409
14.1 Overview...............................................409
14.2 Data Structures .........................................411
14.2.1 Elementary Data Types ............................411
14.2.2 Lists.............................................412
14.2.3 Arrays ...........................................412
14.2.4 Vectors and Matrices ..............................414
14.2.5 Sets .............................................415
14.2.6 Hashes...........................................416
14.3 Compound Statements ...................................417
14.3.1 Conditional Statements ............................417
14.3.2 Loop Statements ..................................417
14.3.3 Keywords break and next .........................419
14.4 Function and Operator Overloading........................419
14.5 Analysis of Algorithms ...................................420
References.....................................................423
Index..........................................................427
