We describe a distributed algorithm for computing the cup product boolean OR : H-1(X, Z) x H-1(X, Z) -> H-2(X, Z) on the cohomology of a finite regular CW-space. A serial implementation of the algorithm is illustrated in two applied topological settings: (i) 3-dimensional digital images; (ii) topological data analysis of a finite sample of points from a metric space. For the second of these illustrations we introduce a cohomological enrichment of the Mapper clustering procedure which may be of independent interest.