  Bouc first studied the topological properties of the matching complex of the
complete graph of order $n$, denoted by $M_n$, in connection with Brown
complexes and Quillen complexes. Later Bj\"{o}rner et al. showed that $M_n$ is
homotopically $(\nu_n-1)$-connected, where
$\nu_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, and conjectured that this connectivity
bound is sharp. Shareshian and Wachs settled the conjecture by inductively
showing that the $\nu_n$-dimensional homology group of $M_n$ is nontrivial,
with Bouc's calculation $H_1(M_7)$ serving as the pivotal base step. In
general, the topology of $M_n$ is not very well-understood, even for a small
$n$. In the present article, we look into the topology of $M_n$, and $M_7$ in
particular, in the light of discrete Morse theory as developed by Forman. We
first construct a gradient vector field on $M_n$ (for $n \ge 5$) that doesn't
admit any critical simplices of dimension up to $\nu_n-1$, except one
unavoidable $0$-simplex, which also leads to the aforementioned
$(\nu_n-1)$-connectedness of $M_n$ 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 $M_7$ 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 $M_7$ in an efficient and
algorithmic manner. We also augment this near-optimal gradient vector field to
one that we conjecture to be optimal.
