On the 3-Torsion Part of the Homology of the Chessboard Complex

Abstract

Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M m, n , which is the simplicial complex of matchings in the complete bipartite graph K m, n . First, we demonstrate that there is nonvanishing 3-torsion inH d (M m, n ; Z) whenever m+n−4 3 ≤ d ≤ m−4 and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d) satisfying H d (M m, n ; Z) = 0. Second, for each k ≥ 0, we show that there is a polynomial f k (a, b) of degree 3k such that the dimension ofH k+a+2b−2 M k+a+3b−1, k+2a+3b−1 ; Z 3 , viewed as a vector space over Z 3 , is at most f k (a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof thatH 2 M 5, 5 ; Z ∼ = Z 3 . Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M m, n to the homology of M m−2, n−1 and M m−2, n−3 .

Cycle-free chessboard complexes and symmetric homology of algebras

“…It is well known that H 2 (∆ 5,5 ) ∼ = Z 3 , [5], [20], [15]. This fact was essentially established in [5], Proposition 2.3.…”

Top homology of hypergraph matching complexes, p-cycle complexes and Quillen complexes of symmetric groups

“…For p = 2, 3 the complexes C p (n) and M p (n) are isomorphic, as for each X ⊆ [n] of size p, there is unique cyclic group of order p in S n having support X. Matching complexes and related complexes have been studied in the literature for their intrinsic combinatorial interest and in connection with applications in various fields of mathematics, see [Wa] for a survey and see [Jo1,Jo2,Jo3,SW] for more recent developments.…”