A constructive approach of Alexander duality

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 \(\mathbb{S}^2\), then each hole of \(X\) corresponds to a connected component of \(\mathbb{S}^2 \backslash X\), and by symmetry, each hole of corresponds to a connected component of \(\mathbb{S}^2 \backslash X\). 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 \(\mathbb{R}^3\).

Some homology quartets, which can be also visualized here:

Aldo Gonzalez-Lorenzo, Alexandra Bac, Yann-Situ Gazull, "A constructive approach of Alexander duality." Journal of Applied and Computational Topology, 2024.
Published Version | HAL Version | Bibtex