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.”