In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality. General statement for spheres Let be a compact, locally contractible subspace of the sphere of dimension n. Let be the complement of in . Then if stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism for all . Note that we can drop local contractibility as part of the hypothesis, if we use Čech cohomology, which is designed to deal with local pathologies. Applications This is useful for computing the cohomology of knot and link complements in . Recall that a knot is an embedding and a link is a disjoint union of knots, such as the Borromean rings. Then, if we write the link/knot as , we have , giving a method for computing the cohomology groups. Then, it is possible to differentiate between different links using the Massey products. For example, for the Borromean rings , the homology groups are Alexander duality for constructible sheaves For smooth manifolds, Alexander duality is a formal consequence of Verdier duality for sheaves of abelian groups. More precisely, if we let denote a smooth manifold and we let be a closed subspace (such as a subspace representing a cycle, or a submanifold) represented by the inclusion , and if is a field, then if is a sheaf of -vector spaces we have the following isomorphism , where the cohomology group on the left is compactly supported cohomology. We can unpack this statement further to get a better understanding of what it means. First, if is the constant sheaf and is a smooth submanifold, then we get , where the cohomology group on the right is local cohomology with support in . Through further reductions, it is possible to identify the homology of with the cohomology of . This is useful in algebraic geometry for computing the cohomology groups of projective varieties, and is exploited for constructing a basis of the Hodge structure of hypersurfaces of degree using the Jacobian ring. Alexander's 1915 result To go back to Alexander's original work, it is assumed that X