Article Dans Une Revue Journal of Applied and Computational Topology Année : 2024

A constructive approach of alexander duality

Résumé

Alexander duality establishes the relation between the homology of an object and the cohomology of its complement in a sphere. For instance, if X is a subset of the 2-dimensional sphere S2, then each hole of X corresponds to a connected component of S2 \ X, and by symmetry, each hole of corresponds to a connected component of S2 \ X. In this paper, we present a new combinatorial and constructive proof of Alexander duality that provides an explicit isomorphism. The proof shows how to compute this isomorphism using a combinatorial tool called the homological discrete vector field. It also provides a one-to-one map between the holes of the object and the holes of its complement, which we use for representing the holes of an object embedded in .

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hal-05003653 , version 1 (18-04-2025)

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Aldo Gonzalez-Lorenzo, Alexandra Bac, Yann-Situ Gazull. A constructive approach of alexander duality. Journal of Applied and Computational Topology, 2024, 9 (1), pp.2. ⟨10.1007/s41468-024-00198-1⟩. ⟨hal-05003653⟩
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