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Cups products in Z2-cohomology of 3D polyhedral complexes


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Opened Access Cups products in Z2-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: 2012
Published in: The Computing Research Repository (CoRR), abs/1207.2346 (2012)
Document type: Article
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.
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