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


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Opened Access 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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