Actually we have found explicit formulas of cup product on 2-dimensional regular CW complexes, whose 2-cells are any polygons, check my thesis (L. Ptackova: A Discrete Wedge Product on Polygonal Pseudomanifolds) or this paper. Our aim has been to create a discrete exterior calculus on general polygonal meshes which we employ for geometry processing tasks. But lately I have been investigating if we can apply our cup product with interesting results in computational topology. So now I am looking for applications of cup products on regular finite CW 2-complexes. By chance, do you know about interesting tasks solvable by cup product in such a low dimension?

Actually we have found explicit formulas of cup product on 2-dimensional regular CW complexes, whose 2-cells are any polygons, check my thesis (L. Ptackova: A Discrete Wedge Product on Polygonal Pseudomanifolds) or this paper. Our aim has been to create a discrete exterior calculus on general polygonal meshes which we employ for geometry processing tasks. But lately I have been investigating if we can apply our cup product with interesting results in computational topology. So now I am looking for applications of cup products on regular finite CW 2-complexes. So I will check how the task of computing cocycles would work, thank you.