  The Morse complex $\mathcal{M}(\Delta)$ of a finite simplicial complex
$\Delta$ is the complex of all gradient vector fields on $\Delta$. In this
paper we study higher connectivity properties of $\mathcal{M}(\Delta)$. For
example, we prove that $\mathcal{M}(\Delta)$ gets arbitrarily highly connected
as the maximum degree of a vertex of $\Delta$ goes to $\infty$, and for
$\Delta$ a graph additionally as the number of edges goes to $\infty$. We also
classify precisely when $\mathcal{M}(\Delta)$ is connected or simply connected.
Our main tool is Bestvina-Brady Morse theory, applied to a "generalized Morse
complex."
