Combinatorial Alexander Duality -- a Short and Elementary Proof

TL;DR

This paper provides a simple, self-contained proof of the combinatorial Alexander duality theorem, which relates the homology of a simplicial complex to the cohomology of its dual.

Contribution

It offers a concise and elementary proof of the combinatorial Alexander duality, making the result more accessible and easier to understand.

Findings

Proves the isomorphism between homology of X and cohomology of X*

Simplifies the proof of a fundamental duality in combinatorial topology

Enhances understanding of duality in simplicial complexes

Abstract

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th 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.