Cups Products in Z2-Cohomology of 3D Polyhedral Complexes
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Let I=(Z^3,26,6,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 Z^3∖ 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);Z_2) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R^3.
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