Higher connectivity of the Morse complex

The Morse complex $\mathcal{M}(\Delta)$ of a finite simplicial complex $\Delta$ is the complex of all gradient vector fields on $\Delta$. In particular $\mathcal{M}(\Delta)$ encodes all possible discrete Morse functions (in the sense of Forman) on $\Delta$. In this paper we find sufficient conditions for $\mathcal{M}(\Delta)$ to be connected or simply connected, in terms of certain measurements on $\Delta$. When $\Delta=\Gamma$ is a graph we get similar sufficient conditions for $\mathcal{M}(\Gamma)$ to be $(m-1)$-connected. The main technique we use is Bestvina-Brady discrete Morse theory, applied to a "generalized Morse complex" that is easier to analyze.