On the complexity of computing the homology type of a triangulation

Abstract

An algorithm for computing the homology type of a triangulation is analyzed. By triangulation is meant a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computer-aided design to tell whether two finite-element meshes or Bezier-spline surfaces are of the same topological type, and whether they can be embedded in R/sup 3/. Homology computation is a pure combinatorial problem of considerable intrinsic interest. While the worst-case bounds obtained for this algorithm are poor, it is argued that many triangulations (in general) and virtually all triangulations in design are very sparse in a particular sense. This sparseness measure is formalized, and a probabilistic analysis of the sparse case is performed to show that the expected running time, of the algorithm is roughly quadratic in the geometric complexity (number of simplices) and linear in the dimension.<>