The cohomology algebra of a space encodes topological information not captured by its cohomology groups. For example, the cohomology groups of S^1 \vee S^1 \vee S^2 and S^2 \times S^2 are isomorphic but their cohomology algebras are not. Classically, one computes the cohomology algebra of a space from the combinatorics of a homeomorphic simplicial complex, but this can be computationally inefficient. In this talk we apply the transfer to merge the simplices in a simplicial complex and obtain a polyhedral complex with fewer cells. Doing so simplifies the combinatorics and improves the computational efficiency while preserving the algebraic topology. We show how to compute cup products directly from the combinatorics of the resulting polyhedral complex and consider some medical applications.