
On the Betti Numbers of Chessboard Complexes
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Abstract:
In this paper we study the Betti numbers of a type of simplicial
complex known as a chessboard complex. We obtain a formula for their Betti
numbers as a sum of terms involving partitions. This formula allows us
to determine which is the first nonvanishing Betti number (aside from
the $0$th Betti number). We can therefore settle certain cases of a
conjecture of \bjetal in \cite{bjorner}. Our formula also shows that all
eigenvalues of the Laplacians of the simplicial complexes are integers,
and it gives a formula (involving partitions)
for the multiplicities of the eigenvalues.
