Computational topology. An introduction.

*(English)*Zbl 1193.55001
Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241 p. (2010).

There are innumerable books available that discuss topology from a purely theoretical point of view. However, there are currently very few that address the practical computational aspects: the data structures used to represent topological spaces, and the algorithms to compute topological information. Computational Topology: An Introduction helps to bridge this gap, and provides a nice overview of computational topology and its applications.

The book is divided into three parts. The first part discusses basic topological and geometric concepts, and gives data structures and algorithms related to these concepts. This portion of the book is somewhat encyclopedic, covering a fair number of topics in a short stretch. The discussion starts with one dimensional figures: graphs and planar curves; proceeds on to surfaces: triangulations, immersions, and mesh simplification; and ends with various types of simplicial complexes: Čech, Vietoris-Rips, Delaunay, and alpha. Proofs of relevant results are occasionally given, but more often they are simply stated.

The second part of the text gives a treatment of the homology of a simplicial complex. Since only coefficients in the field of two elements are considered, the discussion is somewhat simplified in comparison to a standard treatment of homology theory. Although the emphasis is on computation, indeed the matrix reduction algorithm is covered, most of the standard fare is discussed: relative homology, exact sequences, and cohomology. Duality theorems: Poincaré, Lefschetz, and Alexander, are also discussed; and the last chapter of this part of the text covers Morse Theory. The purpose of the discussion is to convey the main ideas, rather than a complete understanding of the details; as with the first part, most results are stated without proof.

The third and final part of the text discusses persistent homology and its applications. Persistent homology is a fairly recent discovery: the original paper was published in 2002, and provides a convenient graphical representation, called a persistence diagram, of the homological information (essentially the Betti numbers) present in a filtration of a topological space. The first chapter in this part of the text discusses the persistent homology of a simplicial complex filtered by ordering its simplices: a data structure and algorithm used for efficient computation, as well as the even more recent (2009) notion of extended persistence of a simplicial manifold. The penultimate chapter discusses how persistence diagrams are affected by changing the ordering of the simplices. The book concludes with a chapter on applications to molecular biology: gene expression and protein docking, as well as applications to biological imaging: cell segmentation and detecting root architecture.

The text is well-written and well-organized, with only a few typographical errors scattered throughout. A broad range of topics are covered, although there is an understandable bias towards topics that reflect the authors’ own interests; indeed some topics in computational topology, most notably Forman’s discrete version of Morse Theory, are not mentioned. While the book is intended as a textbook for graduates and advanced undergraduates in mathematics or computer science, the expert is also likely to find some of the discussions of interest. The exposition is often terse, potentially requiring the less experienced reader to consult other references on some topics; however, the authors provide bibliographic notes at the end of each section. Exercises are also provided at the end of each chapter.

The book is divided into three parts. The first part discusses basic topological and geometric concepts, and gives data structures and algorithms related to these concepts. This portion of the book is somewhat encyclopedic, covering a fair number of topics in a short stretch. The discussion starts with one dimensional figures: graphs and planar curves; proceeds on to surfaces: triangulations, immersions, and mesh simplification; and ends with various types of simplicial complexes: Čech, Vietoris-Rips, Delaunay, and alpha. Proofs of relevant results are occasionally given, but more often they are simply stated.

The second part of the text gives a treatment of the homology of a simplicial complex. Since only coefficients in the field of two elements are considered, the discussion is somewhat simplified in comparison to a standard treatment of homology theory. Although the emphasis is on computation, indeed the matrix reduction algorithm is covered, most of the standard fare is discussed: relative homology, exact sequences, and cohomology. Duality theorems: Poincaré, Lefschetz, and Alexander, are also discussed; and the last chapter of this part of the text covers Morse Theory. The purpose of the discussion is to convey the main ideas, rather than a complete understanding of the details; as with the first part, most results are stated without proof.

The third and final part of the text discusses persistent homology and its applications. Persistent homology is a fairly recent discovery: the original paper was published in 2002, and provides a convenient graphical representation, called a persistence diagram, of the homological information (essentially the Betti numbers) present in a filtration of a topological space. The first chapter in this part of the text discusses the persistent homology of a simplicial complex filtered by ordering its simplices: a data structure and algorithm used for efficient computation, as well as the even more recent (2009) notion of extended persistence of a simplicial manifold. The penultimate chapter discusses how persistence diagrams are affected by changing the ordering of the simplices. The book concludes with a chapter on applications to molecular biology: gene expression and protein docking, as well as applications to biological imaging: cell segmentation and detecting root architecture.

The text is well-written and well-organized, with only a few typographical errors scattered throughout. A broad range of topics are covered, although there is an understandable bias towards topics that reflect the authors’ own interests; indeed some topics in computational topology, most notably Forman’s discrete version of Morse Theory, are not mentioned. While the book is intended as a textbook for graduates and advanced undergraduates in mathematics or computer science, the expert is also likely to find some of the discussions of interest. The exposition is often terse, potentially requiring the less experienced reader to consult other references on some topics; however, the authors provide bibliographic notes at the end of each section. Exercises are also provided at the end of each chapter.

Reviewer: Jason Hanson (Redmond)

##### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |

55-04 | Software, source code, etc. for problems pertaining to algebraic topology |

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55T05 | General theory of spectral sequences in algebraic topology |