HIGHER CONNECTIVITY OF THE MORSE COMPLEX

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