Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering.
The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.
Herbert Edelsbrunner, Duke University, Durham, NC, and Geomagic, Research Triangle Park, NC, and John L. Harer, Duke University, Durham, NC
Preface A Computational Geometric Topology Graphs Connected Components Curves in the Plane Knots and Links Planar Graphs Exercises Surfaces 2-dimensionalManifolds Searching a Triangulation Self-intersections Surface Simplification Exercises Complexes Simplicial Complexes Convex Set Systems Delaunay Complexes Alpha Complexes Exercises B Computational Algebraic Topology Homology Homology Groups Matrix Reduction Relative Homology Exact Sequences Exercises Duality Cohomology Poincar´e Duality Intersection Theory Alexander Duality Exercises Morse Functions Generic Smooth Functions Transversality Piecewise Linear Functions Reeb Graphs Exercises C Computational Persistent Topology Persistence Persistent Homology Efficient Implementations Extended Persistence Spectral Sequences Exercises Stability 1-parameter Families Stability Theorems Length of a Curve Bipartite GraphMatching Exercises Applications Measures for Gene Expression Data Elevation for Protein Docking Persistence for Image Segmentation Homology for Root Architectures Exercises References Index