On Computing the Homology Type of a Triangulation

Abstract

We analyze an algorithm for computing the homology type of a triangulation. By triangulation, we mean 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 $\Re^{3}$. Homology computation is a purely combinatorial problem of considerable intrinsic interest. While the worst-case bounds we obtain for this algorithm are poor, we argue that many triangulations (in general) and virtually all triangulations in design are very "sparse," in a sense we make precise. We formalize this sparseness measure, and perform a probabilistic analysis of the sparse case 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.