Note on a Combinatorial Application of Alexander Duality

✍ Scribed by Anders Björner; Lynne M. Butler; Andrey O. Matveev

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 245 KB
• Volume: 80
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2794

✦ Synopsis

The Mo bius 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 complex 2 that is not a simplex to the cohomology of the complex 1 of nonempty sets of minimal non-faces that do not cover the vertex set of 2.

1997 Academic Press

1. THE MO BIUS NUMBER RESULT

Recall that if P =P _ [0 , 1 ] is a finite poset, the vertices of its incomparability graph G are the elements of P and the edges of G are the 2-element antichains in P. The Mo bius number + P (0 , 1 ) is, by Philip Hall's theorem, the reduced Euler characteristic /~(2) of the order complex article no. TA972794 163 0097-3165Â97 25.00