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Computational homology. (English) Zbl 1039.55001
Applied Mathematical Sciences 157. New York, NY: Springer (ISBN 0-387-40853-3/hbk). xvii, 480 p. (2004).
This book provides the conceptual background for computational homology – a powerful tool used to study the properties of spaces and maps that are insensitive to small perturbations. The material presented here is a unique combination of current research and classical rigor, computation and application. The book is divided into three parts: Part I, which contains the core material on computational homology; Part II, which describes applications and extensions; Part III, which contains a variety of preliminary materials and a brief introduction to topology, abstract algebra, and computer algorithms.
Part I has a Preview on the applications and on the homology of spaces. After a rigorous introduction to the homology of spaces, in the second chapter, it is given a combinatorial approach to the subject: cubical homology (in considering digital images the basic building blocks are pixels identified with squares and for these the algorithms are quite simple). Then the authors present an algorithm based on linear algebra over the integers which demonstrates that the homology group of a topological space made up of cubes is computable. Combinatorial techniques for reducing the number of elements involved in the computation are also presented. The construction of the homology maps is done considering this problem from the viewpoint of multivalued maps. Then the algorithms for the computation of homology maps are discussed. In the second part the authors present the role of algebraic topology in the analysis of numerical and experimental data. In the chapter Prospects in Digital Image Processing the cubical complexes are related to image data and numerically generated data, examples are given in which the authors gain information by computing the homology groups of a cubical set directly related to a numerical image (an efficient computational homology theory based on cubes with potential applications to problems in which it is important to be able to resolve geometric structure). With more sophisticated algebraic concepts and computational techniques related to homological algebra they indicate how homology can be used to study nonlinear dynamics. Combining numerical analysis with homology via computer, they obtain mathematically rigorous results. They also introduce the Wazewski principle and give a straightforward example of how nontrivial topology can be used to draw conclusions about dynamics. In the last chapter of this part they extend the cubical theory to polyhedra and thus place the algorithms developed in the proper context with a the larger field of algebraic topology. This is a bridge between the introductory material of the book and the rich, deep, widely developed field of algebraic topology. Interesting applications require the homology software (by P. Pilarczyk) available at the Computational Homology Program web site, which also contains errata and additional links.

MSC:
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
68T10 Pattern recognition, speech recognition
68U10 Computing methodologies for image processing
37M99 Approximation methods and numerical treatment of dynamical systems
55M25 Degree, winding number
37B10 Symbolic dynamics
37B30 Index theory for dynamical systems, Morse-Conley indices
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