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Constructive Mayer-Vietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes

Dobrina Boltcheva 1 Sara Merino Aceitunos 1 Jean-Claude Léon 1, 2 Franck Hétroy 1
1 EVASION - Virtual environments for animation and image synthesis of natural objects
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
2 G-SCOP_SIREP - Système d’Information, conception RobustE des Produits
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : In this research report, we present an efficient method for computing the homology of a large simplicial complex from the homologies of its sub-complexes. The method uses a constructive version of the Mayer-Vietoris exact sequence which is an algebraic tool relating the homology of a topological space to the homologies of its sub-spaces and their intersection. The method starts by decomposing the input simplicial complex into smaller sub-complexes, for which the homology is computable with the Smith Normal Form reduction algorithm. Then, the method uses the Mayer-Vietoris sequence on the decomposition graph and computes the homology of the input complex by recursive unions of the homological attributes of the sub-complexes. The proposed method outputs all homological attributes (Betti numbers, torsion coefficients and generators) and may be applied to any kind of finite simplicial complexes (manifold/non-manifold, orientable or not, embeddable or not, with heterogeneous dimensionality, etc.)
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https://hal.inria.fr/inria-00542717
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Submitted on : Friday, December 3, 2010 - 12:33:29 PM
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Dobrina Boltcheva, Sara Merino Aceitunos, Jean-Claude Léon, Franck Hétroy. Constructive Mayer-Vietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes. [Research Report] RR-7471, INRIA. 2010. ⟨inria-00542717⟩

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