Complexes of strong discrete Morse matchings

Abstract

Using the strong discrete Morse theory developed by Fernández, we define the complex $\mathcal{SM}(K)$ of strong discrete Morse matchings on a simplicial complex $K$, as well as the pure subcomplex ${\mathcal{SM}}_{pure}(K)$ generated by the facets in $\mathcal{SM}(K)$ of maximal dimension. For most complexes $K$ these objects are proper subcomplexes of the complexes $\mathfrak{M}(K)$ and ${\mathfrak{M}}_{\text{pure}}(K)$ defined by Chari--Joswig using all Morse matchings on $K$. The homotopy types of the latter are not well-understood in general, but they are known when $K$ is the path $P_n$ with $n$ edges, the cycle $C_n$ with $n$ edges, the star $S_n$ with $n$ leaves, the $n$-simplex ${\Delta }^n$ ($n\le 3$), and the boundary $\partial {\Delta }^n$ ($n\le 3$). in this paper we compute the homotopy types of ${\mathcal{SM}}_{pure}(C_n)$ and $\mathcal{SM}(K)$ for $K=P_n,S_n,{\Delta }^n,\partial {\Delta }^n$ for all $n$. We also compute the homology of $\mathcal{SM}(C_n)$ for $n\le 21$.