Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X *={σ⊆V∣V∖σ ∉ X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X * (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.
['Anders Björner', 'Martin Tancer']
Note: Combinatorial Alexander Duality—A Short and Elementary Proof