##  [Alexander Duality](/alexander-duality-0) 

 Definition

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

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

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

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

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

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

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

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

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.