Computational Topology - An Introduction

294 Pages·2008·2.321 MB·English

by  Herbert Edelsbrunner, John Harer

COMPUTATIONAL TOPOLOGY
AN INTRODUCTION
Herbert Edelsbrunner and John Harer
Departments of Computer Science and Mathematics
Duke University

To our families and friends

Table of Contents
Part A
I Graphs 1
1 Connected Components 2
2 Curves in the Plane 9
3 Knots and Links 15
4 Planar Graphs 22
Exercises 29
II Surfaces 31
1 Two-dimensionalManifolds 32
2 Searching a Triangulation 39
3 Self-intersections 45
4 Surface Simplification 51
Exercises 57
III Complexes 59
1 Simplicial Complexes 60
2 Convex Set Systems 67
3 Delaunay Complexes 74
4 Alpha Complexes 81
Exercises 88
Part B
IV Homology 91
1 Homology Groups 92
2 Matrix Reduction 98
3 Relative Homology 104
4 Exact Sequences 111
Exercises 118
V Duality 121
1 Cohomology 122
2 Poincar´e Duality 128
3 Intersection Theory 132
4 Alexander Duality 136
Exercises 139
VI Morse Functions 141
1 Generic Smooth Functions 142
2 Transversality 148
3 Piecewise Linear Functions 154
4 Reeb Graphs 160
Exercises 167
Part C
VII Persistence 169
1 Persistent Homology 170
2 Efficient Implementations 178
3 Extended Persistence 184
4 Spectral Sequences 191
Exercises 198
VIII Stability 201
1 Time Series 202
2 Stability Theorems 208
3 Length of a Curve 215
4 Bipartite Graph Matching 221
Exercises 228
IX Applications 231
1 Image Segmentation
2 Elevation
3 Gene Expression
4 Local Homology for Plant Root Architecture
Exercises
Part D
X Open Problems 237
1 Complexity of Reidemeister Moves 238
2 Shelling a 3-ball 239
3 Geometric Realization of 2-manifolds 241
4 Embedding in Three Dimensions 242
5 Equipartion in Four Dimensions 243
6 Running-time of Matrix Reduction 244
7 Multi-parameter Persistence 245
8 Unfolding PL Critical Points 246
9 PL in the Limit 247
10 Counting Halving Sets 248
Index 259
Preface
The last ten years have witnessed that geometry, topology, and algorithms
form a potent mix of disciplines with many applications inside and outside
academia. We aim at bringing these developments to a larger audience. This
book has been written to be taught, andit is based onnotes developed during
courses delivered at Duke University and at the Berlin Mathematical School,
primarily to students of computer science and mathematics. The organization
into chapters,sections,exercises,andopenproblemsreflects the teaching style
we practice. Each chapter develops a major topic and is worth about two
weeks of teaching. The chapters are divided into sections, each a lecture of
one and a quarter hours. To convey a feeling for the boundary of the current
knowledge, we complement the material with descriptions of open problems.
An interesting challenge is the mixed background of the audience. How do
we teach topology to students with limited background in mathematics, and
howdoweconveyalgorithmstostudentswithlimitedbackgroundincomputer
science? Assuming no prior knowledgeand appealing to the intelligence of the
listenerisagoodfirststep. Motivatingthe materialbyrelatingittosituations
indifferentwalksoflife is helpfulinbuilding upintuitionthatcancutthrough
otherwisenecessaryformalism. Exposingcentralideaswithsimplemeanshelps
and so does minimizing the necessary amount of detail.
The material in this book is a combination of topics in geometry, topology,
and algorithms. Far from getting diluted, we find that the fields benefit from
each other. Geometry gives a concrete face to topological structures and al-
gorithms offer a means to construct them at a level of complexity that passes
the threshold necessary for practical applications. As always, algorithms have
to be fast because time is the one fundamental resource human kind did not
yetlearnto manipulate forits selfishpurposes. Beyondthese obviousrelation-
ships, there is a symbiotic affinity between algorithms and the algebra used to
capture topological information. It is telling that both fields trace their name
back to the writing of the same Persian mathematician, al-Khwarizmi, work-
ing in Baghdad during the ninth century after Christ. Besides living in the
triangle spanned by geometry, topology, and algorithms, we find it useful to
contemplate the place of the material in the tension between extremes such as
local vs. global;
discrete vs. continuous;
abstract vs. concrete;
intrinsic vs. extrinsic.
Global insights are often obtained by a meaningful integration of local infor-
mation. This is how we proceed in many fields, taking on bigger challenges
after mastering the small. But small things are big from up close and big ones
small from afar. Indeed, the question of scale lurking behind this thought is
the driving force for much of the development described in this book. The
dichotomy between discrete and continuous structures is driven by opposing
goals,machinecomputationandhumanunderstanding. Bothareillusionsthat
are useful to have but should not be confused with anything intangible like
reality. The tension between the abstract and the concrete as well as between
the intrinsic and the extrinsic have everything to do with human approach to
knowledge. An example close to home is the step from geometry to topology
in which we remove the burdens of size to focus on the phenomenon of con-
nectivity. The more abstract the context the more general the insight. Now,
generality is good but it is not a substitute for the concrete steps that have
to be taken to build bridges to applications. Zooming in and out of generality
leads to unifying viewpoints and suggests meaningful integrations where they
exist.
While these thoughtshavecertainlyinfluencedus inthe selectionofthe ma-
terial and in its presentation, there is a long way to the concrete instantiation
we call this book. The hardest partis to land, and we do in four parts decom-
posedintoatotaloftenchapters. PartAisagentleintroductiontotopological
thought. Discussing Graphs in Chapter I, Surfaces in Chapter II, and Com-
plexes in Chapter III, we gradually build up topologicalsophistication, always
incombinationwithgeometricandalgorithmicideas. PartBpresentsclassical
materialfrom topology. We focus onwhat we deem useful andefficiently com-
putable. The material on Homology in Chapter IV and on Duality in Chapter
V is exclusively algebraic. In the discussion of Morse Theory in Chapter VI,
we build a bridge to differential concepts in topology. Part C is mostly novel
and indeed the main reason we write this book. The main new concept is
Persistence introduced in Chapter VII and its Stability discussed in Chapter
VIII. Finally, we address connections to Singularities in Chapter IX. Part D
concludes the book with a small collection of open problems in computational