Cups products in Z2-cohomology of 3D polyhedral complexes

Abstract

Let I=(Z3,26,6,B) be a 3D digital image, let Q(I) be the associated cubical complex and let ∂Q(I) be the subcomplex of Q(I) whose maximal cells are the quadrangles of Q(I) shared by a voxel of B in the foreground -- the object under study -- and by a voxel of Z3∖B in the background -- the ambient space. We show how to simplify the combinatorial structure of ∂Q(I) and obtain a 3D polyhedral complex P(I) homeomorphic to ∂Q(I) but with fewer cells. We introduce an algorithm that computes cup products on H∗(P(I);Z2) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R3.