Alexander duality

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 P. S. 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.

Let X be a compact, locally contractible subspace of the sphere S of dimension n. Let Y be the complement of X in S. Then if H stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism between

and

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.

To go back to Alexander's original work, it is assumed that X is a simplicial complex.

Alexander had little of the modern apparatus, and his result was only for the Betti numbers, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open if the other is closed, but this doesn't affect its homology. Each of the solid tori is from the homotopy point of view a circle. If we just write down the Betti numbers

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