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Computing Cup Products in \(\mathbb {Z}_2\)-Cohomology of 3D Polyhedral Complexes

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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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Notes

  1. http://grupo.us.es/cimagroup/imagesequence2cupproduct.zip.

  2. http://www.mathworks.com/matlabcentral/fileexchange/2762-dicom-example-files/.

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Acknowledgments

This research was funded in part by the Spanish Ministry of Economy and Competitiveness under Project MTM2012-32706 and a Millersville University faculty research grant. We wish to thank Jim Stasheff and the anonymous referees for their helpful suggestions, which significantly improved the exposition, and Manuel Eugenio Herrera Lara–Universidad Complutense de Madrid for providing the 14 micro-CT images in Fig. 5.

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Authors and Affiliations

  1. Department of Applied Mathematics (I), School of Computer Engineering, University of Seville, Campus Reina Mercedes, C.P. 41012 , Seville, Spain

    Rocio Gonzalez-Diaz

  2. Pattern Recognition Department, Advanced Technologies Application Center, 7th Avenue #21812 218 and 222, Siboney, Playa, C.P. 12200 , Havana, Cuba

    Javier Lamar

  3. Department of Mathematics, Millersville University of Pennsylvania, P.O. Box 1002, Millersville, PA , 17551-0302, USA

    Ronald Umble

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  1. Rocio Gonzalez-Diaz
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Correspondence to Rocio Gonzalez-Diaz.

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Communicated by Herbert Edelsbrunner.

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Gonzalez-Diaz, R., Lamar, J. & Umble, R. Computing Cup Products in \(\mathbb {Z}_2\)-Cohomology of 3D Polyhedral Complexes. Found Comput Math 14, 721–744 (2014). https://doi.org/10.1007/s10208-014-9193-0

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