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

TL;DR

This paper investigates the homotopy types and structural properties of complexes of discrete Morse matchings on simplices, providing explicit computations for low dimensions and general formulas linking matchings to skeletons.

Contribution

It computes homotopy types for specific complexes of discrete Morse matchings and establishes a reduction formula for enumerating optimal matchings in higher dimensions.

Findings

Homotopy types of complexes for $ abla^3$ and $ abla^4$ are explicitly computed.

A formula relating the number of top-dimensional facets to the enumeration of matchings is established.

The $f$-vector of $ abla^4$ is computed, with 380,125 optimal matchings.

Abstract

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 $n$.