Cup Products in Computational Topology

Topological persistence methods provide a robust framework for analyzing large point cloud datasets topologically, and have been applied with great success towards homology computations on simplicial complexes. In this paper, we apply the persistence algorithm towards calculating a set of invariants related to the cup product structure on the cohomology ring for a space. These invariants express the extent to which cohomology classes can be obtained as cup products of lower dimensional cohomology classes in the space. To calculate these invariants, we apply persistence to a chain complex associated with the Alexander-Whitney product in homology which dualizes to be the cup product in cohomology. We show that the method is practical to implement by showing results from an implementation for PLEX, a Matlab toolkit which was developed for such topological computations.