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On the 3-Torsion Part of the Homology of the Chessboard Complex
Jakob Jonsson
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
jakobj@math.kth.se
Annals of Combinatorics 14 (4) pp.487-505 December, 2010
AMS Subject Classification: 55U10, 05E25
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 in d(M[m,n];Z) whenever dm-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 d(M[m,n];Z) ≠ 0. Second, for each k ≥ 0, we show that there is a polynomial fk(a,b) of degree 3k such that the dimension of {k+a+2b-2}(M[k+a+3b-1,k+2a+3b-1];Z3), viewed as a vector space over Z3, is at most fk(a,b) for all a ≥ 0 and b ≥ k+2. Third, we give a computer-free proof that 2(M[5,5];Z) ≅ Z3. 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].
Keywords: matching complex, chessboard complex, simplicial homology

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