@article{gonzalez-lorenzo2024-constructive_alexander_duality,
    author = "Gonzalez-Lorenzo, Aldo and Bac, Alexandra and Gazull, Yann-Situ",
    title = "A constructive approach of {A}lexander duality",
    journal = "Journal of Applied and Computational Topology",
    year = "2024",
    month = "12",
    number = "2",
    volume = "9",
    url = "https://doi.org/10.1007/s41468-024-00198-1",
    doi = "10.1007/s41468-024-00198-1",
    abstract = "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 R3."
}