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# Computing cup products in $$\mathbb {Z}_2$$-cohomology of 3D polyhedral complexes

Computing cup products in $$\mathbb {Z}_2$$-cohomology of 3D polyhedral complexes
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 Author: González Díaz, Rocío Lamar León, Javier Umble, Ronald Department: Universidad de Sevilla. Departamento de Matemática Aplicada I Date: 2014 Published in: Foundations of Computational Mathematics, 14 (4), 721-744. Document type: Article Abstract: Let $$I=(\mathbb {Z}^3,26,6,B)$$ be a three-dimensional (3D) digital image, let $$Q(I)$$ be an associated cubical complex, and let $$\partial Q(I)$$ be a 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 $$\mathbb {Z}^3\backslash B$$ in the background—the ambient space. We show how to simplify the combinatorial structure of $$\partial Q(I)$$ and obtain a 3D polyhedral complex $$P(I)$$ homeomorphic to $$\partial Q(I)$$ but with fewer cells. We introduce an algorithm that computes cup products in $$H^*(P(I);\mathbb {Z}_2)$$ directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in $$\mathbb {R}^3$$.
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DOI: http://dx.doi.org/10.1007/s10208-014-9193-0

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