Computational Homology [Hardcover]

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.

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.

PrefacePart I Homology1 Preview1.1 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 ColÄ