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Bouc (1992) first studied the topological properties of Mn, the matching complex of the complete graph of order n, in connection with Brown complexes and Quillen complexes. Björner et al. (1994) showed that Mn is homotopically (νn−1)-connected, where νn=⌊n+13⌋−1, and conjectured that this connectivity bound is sharp. Shareshian and Wachs (2007) settled the conjecture by inductively showing that the νn-dimensional homology group of Mn is nontrivial, with Bouc's calculation of H1(M7) serving as the pivotal base step. In general, the topology of Mn is not very well-understood, even for a small n. In the present article, we look into the topology of Mn, and M7 in particular, in the light of discrete Morse theory as developed by Forman (1998). We first construct a gradient vector field on Mn (for n≥5) that doesn't admit any critical simplices of dimension up to νn−1, except one unavoidable 0-simplex, which also leads to the aforementioned (νn−1)-connectedness of Mn in a purely combinatorial way. However, for an efficient homology computation by discrete Morse theoretic techniques, we are required to work with a gradient vector field that admits a low number of critical simplices, and also allows an efficient enumeration of gradient paths. An optimal gradient vector field is one with the least number of critical simplices, but the problem of finding an optimal gradient vector field, in general, is an NP-hard problem (even for 2-dimensional complexes). We improve the gradient vector field constructed on M7 in particular to a much more efficient (near-optimal) one, and then with the help of this improved gradient vector field, compute the homology groups of M7 in an efficient and algorithmic manner. We also augment this near-optimal gradient vector field to one that we conjecture to be optimal. |
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