A constructive approach of alexander dualityA constructive approach...A. Gonzalez-Lorenzo et al.

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2$$\end{document}, then each hole of X corresponds to a connected component of S2\X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2 \setminus X$$\end{document}, and by symmetry, each hole of S2\X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2 \setminus X$$\end{document} corresponds to a connected component of 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}.