Chessboard Complexes and Matching Complexes

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“…This was indeed verified by Xun Dong (personal communication) for γ,δ when γ has at most 2 parts. However Dong points out that it is not true already for the 3 × 4 chessboard complex (1,1,1),(1,1,1,1) , since it was observed in [5] that this complex triangulates a 2-dimensional torus.…”

“…By a result of Gay [13], this is an instance of the plethysm problem [12 Hashimoto [14] was the first to show that Tor A 5,5 3 (Segre(5, 5, 0), k) depends upon whether k has characteristic 3, and consequently that 5,5 has 3-torsion in its 2-homology (see also [5,Proposition 2.3] which contains an error that was later corrected). Anderson [3] showed that Tor A 7 5 (Veronese(7, 2, 0), k) depends upon whether k has characteristic 5, by an explicit calculation ofH 4 ( γ , Z) for the multidegree γ = (2, 2, 2, 2, 2, 2, 2).…”

“…. , 1), this complex γ is the matching complex n for a complete graph on n vertices, as considered in [5]. The matching complex for a graph G is the simplicial complex whose vertex set is the set of edges of G, and whose simplices are the subsets of edges which form a partial matching, i.e.…”

“…. , 1)), this complex γ,δ = m,n is the m × n chessboard complex considered in [5], whose vertices are the squares of the chessboard, and whose simplices are the sets of squares which correspond to a placement of rooks on the board so that no two rooks lie in the same row or column. The complex 3,3 is depicted in figure 4(a).…”

“…The observation says that in the case where M is a finitely generated semigroup module over an affine semigroup ring S, and A is the polynomial ring in the generators of S, the groups Tor Ȧ (M, k) are isomorphic to direct sums of homology groups with coefficients in k for certain simplicial complexes derived from S, M. As will be shown in Section 3 (and was alluded to briefly in [7]), this result applies to both Segre(m, n, r ) and Veronese(n, 2, r ). Furthermore, the relevant simplicial complexes include as special cases the m × n chessboard complexes m,n and the matching complex n for the complete graph on n-vertices, as defined and studied in [5]. Our computations of Tor allow us to compute the rational homology (Theorem 3.3) for all chessboard complexes with multiplicities, as defined in [7,Remark 3.5], and for the class of complexes generalizing the matching complexes n which we call bounded-degree graph complexes.…”