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 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 formula_1 be a compact, locally contractible subspace of the sphere formula_2 of dimension "n". Let formula_3 be the complement of formula_1 in formula_2. Then if formula_6 stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism
for all formula_8. 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 formula_9. Recall that a knot is an embedding formula_10 and a link is a disjoint union of knots, such as the Borromean rings. Then, if we write the link/knot as formula_11, we have
giving a method for computing the cohomology groups. Then, it is possible to differentiate between different links using the Massey products. The homology groups are
for formula_11 the Borromean rings.
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 formula_1 denote a smooth manifold and we let formula_16 be a closed subspace (such as a subspace representing a cycle, or a submanifold) represented by the inclusion formula_17, and if formula_18 is a field, then if formula_19 is a sheaf of formula_18-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 formula_22 is the constant sheaf and formula_23 is a smooth submanifold, then we get
where the cohomology group on the right is local cohomology with support in formula_23. Through further reductions, it is possible to identify the homology of formula_26 with the cohomology of formula_23. 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 formula_28 using the Jacobian ring.
Alexander's 1915 result.
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 does not 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
of the circle (up to formula_29, since we are in the 3-sphere), then reverse as
and then shift one to the left to get
there is a difficulty, since we are not getting what we started with. On the other hand the same procedure applied to the "reduced" Betti numbers, for which the initial Betti number is decremented by 1, starts with
and gives
whence
This "does" work out, predicting the complement's reduced Betti numbers.
The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers
of the circle, and therefore
by flipping over and
by shifting to the left. This gives back something different from what the Jordan theorem states, which is that there are two components, each contractible (Schoenflies theorem, to be accurate about what is used here). That is, the correct answer in honest Betti numbers is 
Once more, it is the reduced Betti numbers that work out. With those, we begin with
to finish with
From these two examples, therefore, Alexander's formulation can be inferred: reduced Betti numbers formula_30 are related in complements by