The complex of discrete Morse matchings of the $n$-simplex: homotopy types and structural results

The complex of discrete Morse matchings $\M(K)$, introduced by Chari and Joswig, is a simplicial complex whose simplices are the acyclic matchings on the Hasse diagram of $K$. Its homotopy type is known in only a handful of cases. In this paper, we compute the homotopy types of $\M(\Delta^3)$ and $\M(\partial\Delta^3)$, the corresponding pure complexes $\M_{P}(\Delta^3) \simeq \M_{P}(\partial\Delta^3)$, and the generalized complex of discrete Morse matchings $\GM(\Delta^3) \simeq \GM(\partial\Delta^3)$. For general $n$ we prove the identity $f(n) = (n+1) \cdot |\text{top-dimensional facets of } \M(\Delta^n_{(n-2)})|$, reducing the enumeration of optimal matchings on $\Delta^n$ to an enumeration on its $(n-2)$-skeleton, and we show that the inclusion $\M(K) \hookrightarrow \M(CK)$ is null-homotopic for any cone. We also compute the $f$-vector of $\M(\Delta^4)$, whose top entry $f(4) = 380{,}125$ is the number of optimal discrete Morse matchings on $\Delta^4$. We conclude with two conjectures extending the $\M_{P}$ and $\GM$ equivalences to all $n$.