Note on a Combinatorial Application of Alexander Duality

The Möbius number of a finite partially ordered set equals (up to sign) the difference between the number of even and odd edge covers of its incomparability graph. We use Alexander duality and the nerve lemma of algebraic topology to obtain a stronger result. It relates the homology of a finite simplicial complexΔthat is not a simplex to the cohomology of the complexΓof nonempty sets of minimal non-faces that do not cover the vertex set ofΔ