Existence and construction of a cup product for generic polytopal meshes

Determine whether a cup product operation can be defined for generic polytopal meshes (cell complexes whose cells are simple polytopes), and, if so, construct such a family of bilinear maps \smile_{p,q}: C^p\mathcal{K} × C^q\mathcal{K} → C^{p+q}\mathcal{K} that is local with respect to cell boundaries and satisfies graded commutativity and the graded Leibniz rule with respect to the coboundary operator.

Background

In the paper a discrete analogue of the wedge product, called the cup product, is introduced for quasi-cubical meshes and used to formulate variational models on combinatorial meshes. The cup product is defined as a local bilinear operation on cochains that obeys graded commutativity and a graded Leibniz rule relative to the coboundary operator.

The authors note that cup products are known for specific mesh types, such as simplicial, cubical, and certain polygonal meshes, and they leverage a Forman subdivision to obtain a quasi-cubical structure when needed. However, for generic polytopal meshes (not restricted to these special classes), the existence of a suitable cup product remains uncertain, motivating an explicit open problem about its existence and construction under the same structural properties.