A duality theorem that relates the reduced homology groups of a compact subset A of the sphere S^n to the reduced cohomology groups of its complement S^n ackslash A, typically expressed by an isomorphism H~_i(A) ≅ H~^{n-i-1}(S^n ackslash A).

Principle

Homological information about a compact subset in a sphere is encoded dually in the cohomology of its complement; removing A from the sphere exchanges homological dimensions in a predictable way determined by the sphere's dimension.

Demonstration

If A is a finite union of points in S^2, then the reduced homology H~_0(A) (counting components of A minus one) corresponds under Alexander duality to H~^{1}(S^2 ackslash A), so the complement has cohomology in degree one corresponding to linking cycles around those points; this gives concrete computations of complements of knots or point sets.

Misapplication

Using Alexander duality for subsets that are not compact in the sphere, or forgetting to use reduced (co)homology; applying the formula naively in Euclidean space without passing to one-point compactification can produce incorrect degree shifts.

Consequence

Provides an effective computational tool: one can compute invariants of complements (important in knot theory and complement topology) from invariants of the subset itself, and vice versa, giving constraints on possible embeddings and linking phenomena.

Reversal

Viewed in reverse, cohomology classes of the complement detect homology classes of the subset; Alexander duality can be used to deduce properties of A from knowledge of S^n ackslash A, turning complement calculations into direct information about A.

Boundary

Holds for compact subsets of S^n (or locally compact subsets of R^n after one-point compactification) and uses reduced (co)homology; it does not apply without modification to arbitrary noncompact subsets or to non-spherical ambient manifolds without additional hypotheses (Lefschetz or Poincaré duality frameworks may be required).

Semantic Tension

Often compared with Poincaré and Lefschetz dualities; the tension is that Alexander duality is about subset ↔ complement in a sphere, while Poincaré/Lefschetz dualities concern a manifold's homology/cohomology as a whole, so the theorems are related but apply in different contexts and use different hypotheses.

Synthesis

Alexander Duality asserts that homological data of a compact subset of a sphere and cohomological data of its complement are dual: computing one side yields the other via a predictable degree shift tied to the ambient sphere dimension, a principle widely used in complement and embedding problems.