Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.
Author(s): Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek,
Edition: 1
Year: 2004
Language: English
Pages: 499
Contents......Page 14
Preface......Page 8
Part I: Homology......Page 20
1.1 Analyzing Images......Page 22
1.2 Nonlinear Dynamics......Page 32
1.3 Graphs......Page 36
1.4 Topological and Algebraic Boundaries......Page 38
1.5 Keeping Track of Directions......Page 43
1.6 Mod 2 Homology of Graphs......Page 45
2.1 Cubical Sets......Page 58
2.2 The Algebra of Cubical Sets......Page 66
2.3 Connected Components and H[sub(o)](X)......Page 85
2.4 Elementary Collapses......Page 89
2.5 Acyclic Cubical Spaces......Page 98
2.6 Homology of Abstract Chain Complexes......Page 104
2.7 Reduced Homology......Page 107
2.8 Bibliographical Remarks......Page 110
3 Computing Homology Groups......Page 112
3.1 Matrix Algebra over Z......Page 113
3.2 Row Echelon Form......Page 126
3.3 Smith Normal Form......Page 136
3.4 Structure of Abelian Groups......Page 144
3.5 Computing Homology Groups......Page 151
3.6 Computing Homology of Cubical Sets......Page 153
3.7 Preboundary of a Cycle—Algebraic Approach......Page 158
3.8 Bibliographical Remarks......Page 160
4.1 Chain Maps......Page 162
4.2 Chain Homotopy......Page 168
4.3 Internal Elementary Reductions......Page 174
4.4 CCR Algorithm......Page 184
4.5 Bibliographical Remarks......Page 190
5 Preview of Maps......Page 192
5.1 Rational Functions and Interval Arithmetic......Page 193
5.2 Maps on an Interval......Page 195
5.3 Constructing Chain Selectors......Page 204
5.4 Maps of Γ[sup(1)]......Page 208
6.1 Representable Sets......Page 218
6.2 Cubical Multivalued Maps......Page 225
6.3 Chain Selectors......Page 229
6.4 Homology of Continuous Maps......Page 234
6.5 Homotopy Invariance......Page 250
6.6 Bibliographical Remarks......Page 253
7 Computing Homology of Maps......Page 254
7.1 Producing Multivalued Representation......Page 255
7.2 Chain Selector Algorithm......Page 259
7.3 Computing Homology of Maps......Page 261
7.4 Geometric Preboundary Algorithm (optional section)......Page 263
7.5 Bibliographical Remarks......Page 272
Part II: Extensions......Page 274
8.1 Images and Cubical Sets......Page 276
8.2 Patterns from Cahn–Hilliard......Page 278
8.3 Complicated Time-Dependent Patterns......Page 285
8.4 Size Function......Page 288
8.5 Bibliographical Remarks......Page 296
9.1 Relative Homology......Page 298
9.2 Exact Sequences......Page 308
9.3 The Connecting Homomorphism......Page 311
9.4 Mayer–Vietoris Sequence......Page 318
9.5 Weak Boundaries......Page 322
9.6 Bibliographical Remarks......Page 325
10 Nonlinear Dynamics......Page 326
10.1 Maps and Symbolic Dynamics......Page 327
10.2 Differential Equations and Flows......Page 337
10.3 Wazewski Principle......Page 339
10.4 Fixed-Point Theorems......Page 343
10.5 Degree Theory......Page 351
10.6 Complicated Dynamics......Page 361
10.7 Computing Chaotic Dynamics......Page 380
10.8 Bibliographical Remarks......Page 394
11 Homology of Topological Polyhedra......Page 396
11.1 Simplicial Homology......Page 397
11.2 Comparison of Cubical and Simplicial Complexes......Page 404
11.3 Homology Functor......Page 407
11.4 Bibliographical Remarks......Page 412
Part III: Tools from Topology and Algebra......Page 414
12.1 Norms and Metrics in R[sup(d)]......Page 416
12.2 Topology......Page 421
12.3 Continuous Maps......Page 426
12.4 Connectedness......Page 430
12.5 Limits and Compactness......Page 434
13.1 Abelian Groups......Page 438
13.2 Fields and Vector Spaces......Page 446
13.3 Homomorphisms......Page 452
13.4 Free Abelian Groups......Page 460
14.1 Overview......Page 470
14.2 Data Structures......Page 472
14.3 Compound Statements......Page 478
14.4 Function and Operator Overloading......Page 480
14.5 Analysis of Algorithms......Page 481
References......Page 484
F......Page 490
S......Page 491
Z......Page 492
C......Page 494
D......Page 495
H......Page 496
M......Page 497
R......Page 498
Z......Page 499
