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 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.”