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