Note: Combinatorial Alexander Duality-A Short and Elementary Proof

摘要

Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X* = {sigma subset of V vertical bar V sigma is not an element of X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (vertical bar V vertical bar - 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.