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

The complex of discrete Morse matchings $\text{M}(K)$, 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 $\text{M}({\mathrm{\Delta }}^3)$ and $\text{M}(\mathrm{\partial }{\mathrm{\Delta }}^3)$, the corresponding pure complexes ${\text{M}}_P({\mathrm{\Delta }}^3)\simeq {\text{M}}_P(\mathrm{\partial }{\mathrm{\Delta }}^3)$, and the generalized complex of discrete Morse matchings $\text{GM}({\mathrm{\Delta }}^3)\simeq \text{GM}(\mathrm{\partial }{\mathrm{\Delta }}^3)$. For general $n$ we prove the identity $f(n)=(n+1)\cdot \mathrm{\mid }\text{top-dimensional facets of }\text{M}({\mathrm{\Delta }}_{(n-2)}^n)\mathrm{\mid }$, reducing the enumeration of optimal matchings on ${\mathrm{\Delta }}^n$ to an enumeration on its $(n-2)$-skeleton, and we show that the inclusion $\text{M}(K)\hookrightarrow \text{M}(CK)$ is null-homotopic for any cone. We also compute the $f$-vector of $\text{M}({\mathrm{\Delta }}^4)$, whose top entry $f(4)=\mathrm{380,125}$ is the number of optimal discrete Morse matchings on ${\mathrm{\Delta }}^4$. We conclude with two conjectures extending the ${\text{M}}_P$ and $\text{GM}$ equivalences to all $n$.