Higher connectivity of the Morse complex

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