Computational homology / Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek.

Includes bibliographical references (p. [465]-469) and indexes.

.1 Analyzing Images1.2 Nonlinear Dynamics1.3 Graphs1.4 Topological and Algebraic Boundaries1.5 Keeping Track of Directions1.6 Mod 2 Homology of Graphs2 Cubical Homology2.1 Cubical Sets2.1.1 Elementary Cubes2.1.2 Cubical Sets2.1.3 Elementary Cells2.2 The Algebra of Cubical Sets2.2.1 Cubical Chains2.2.2 Cubical Chains in a Cubical Set2.2.3 The Boundary Operator2.2.4 Homology of Cubical Sets2.3 Connected Components and H0(X)2.4 Elementary Collapses2.5 Acyclic Cubical Spaces2.6 Homology of Abstract Chain Complexes2.7 Reduced Homology2.8 Bibliographical Remarks3 Computing Homology Groups3.1 Matrix Algebra over Z3.2 Row Echelon Form3.3 Smith Normal Form3.4 Structure of Abelian Groups3.5 Computing Homology Groups3.6 Computing Homology of Cubical Sets3.7 Preboundary of a Cycle-Algebraic Approach3.8 Bibliographical Remarks4 Chain Maps and Reduction Algorithms4.1 Chain Maps4.2 Chain Homotopy4.3 Internal Elementary Reductions4.3.1 Elementary Collapses Revisited4.3.2 Generalization of Elementary Collapses4.4 CCR Algorithm4.5 Bibliographical Remarks5 PreviewofMaps5.1 Rational Functions and Interval Arithmetic5.2 Maps on an Interval5.3 Constructing Chain Selectors5.4 Maps of A16 Homology of Maps6.1 Representable Sets6.2 Cubical Multivalued Maps6.3 Chain Selectors6.4 Homology of Continuous Maps6.4.1 Cubical Representations6.4.2 Rescaling6.5 Homotopy Invariance6.6 Bibliographical Remarks7 Computing Homology of Maps7.1 Producing Multivalued Representation7.2 Chain Selector Algorithm7.3 Computing Homology of Maps7.4 Geometric Preboundary Algorithm (optional section)7.5 Bibliographical RemarksPart II Extensions8 Prospects in Digital Image Processing8.1 Images and Cubical Sets8.2 Patterns from Cahn-Hilliard8.3 Complicated Time-Dependent Patterns8.4 Size Function8.5 Bibliographical Remarks9 Homological Algebra9.1 Relative Homology9.1.1 Relative Homology Groups9.1.2 Maps in Relative Homology9.2 Exact Sequences9.3 The Connecting Homomorphism9.4 Mayer-Vietoris Sequence9.5 Weak Boundaries9.6 Bibliographical Remarks10 Nonlinear Dynamics10.1 Maps and Symbolic Dynamics10.2 Differential Equations and Flows10.3 Wayzewski Principle10.4 Fixed-Point Theorems10.4.1 Fixed Points in the Unit Ball10.4.2 The Lefschetz Fixed-Point Theorem10.5 Degree Theory10.5.1 Degree on Spheres10.5.2 Topological Degree10.6 Complicated Dynamics10.6.1 Index Pairs and Index Map10.6.2 Topological Conjugacy10.7 Computing Chaotic Dynamics10.8 Bibliographical Remarks11 Homology of Topological Polyhedra11.1 Simplicial Homology11.2 Comparison of Cubical and Simplicial Complexes11.3 Homology Functor11.3.1 Category of Cubical Sets11.3.2 Connected Simple Systems11.4 Bibliographical RemarksPart III Tools from Topology and Algebra12 Topology12.1 Norms and Metrics in Rd12.2 Topology12.3 Continuous Maps12.4 Connectedness12.5 Limits and Compactness13 Algebra13.1 Abelian Groups13.1.1 Algebraic Operations13.1.2 Groups13.1.3 Cyclic Groups and Torsion Subgroup13.1.4 Quotient Groups13.1.5 Direct Sums13.2 Fields and Vector Spaces13.2.1 Fields13.2.2 Vector Spaces13.2.3 Linear Combinations and Bases13.3 Homomorphisms13.3.1 Homomorphisms of Groups13.3.2 Linear Maps13.3.3 Matrix Algebra13.4 Free Abelian Groups13.4.1 Bases in Groups13.4.2 Subgroups of Free Groups13.4.3 Homomorphisms of Free Groups14 Syntax of Algorithms14.1 Overview14.2 Data Structures14.2.1 Elementary Data Types14.2.2 Lists14.2.3 Arrays14.2.4 Vectors and Matrices14.2.5 Sets