On the Betti Numbers of Chessboard Complexes

Authors:
Joel Friedman;Phil Hanlon
Affiliations:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. E-mail: jf@math.ubc.ca;Department of Mathematics, University of Michigan, Ann Arbor, MI 48109–1003. E-mail: hanlon@math.lsa.umich.ed u
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
1998

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Abstract

In this paper we study the Betti numbers of a type of simplicialcomplex known as a chessboard complex. We obtain a formula for their Bettinumbers as a sum of terms involving partitions. This formula allows usto determine which is the first nonvanishing Betti number (aside fromthe 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Živaljević in [2]. Our formula also shows that alleigenvalues of the Laplacians of the simplicial complexes are integers,and it gives a formula (involving partitions)for the multiplicities of the eigenvalues.