Abstract:

Abstract: For $1 \le m \le n$, let $M_{m,n}$ be the family of sets of non-attacking rooks on an $m \times n$ chessboard. Equivalently, a set of rooks belongs to $M_{m,n}$ if and only if there is at most one rook in each row and at most one rook in each column. We may view $M_{m,n}$ as an abstract simplicial complex and thus examine its homology. Assume that $m \le n \le 2m-5$. Shareshian and Wachs proved that the bottom nonvanishing reduced integral homology group of $M_{m,n}$ appears in degree $\lceil \frac{m+n-4}{3}\rceil$, thereby settling a conjecture due to Björner, Lovász, Vre\īcica, and $\check{\mathrm{Z}}$ivaljevi\īc. The goal of the talk is to give an outline of the proof. In particular, we explain how Shareshian and Wachs used exact sequences to detect torsion in the relevant homology group, which is finite for almost all $m$ and $n$. We also show how to extend their result to homology groups of higher degree.