Simplifying Weinstein Morse functions

TL;DR

This paper establishes bounds on the number of critical points for Weinstein Morse functions, linking them to smooth Morse functions, and explores implications for symplectic topology and Legendrian invariants.

Contribution

It proves that Weinstein Morse functions can be simplified to near minimal critical points, providing new bounds and topological obstructions in symplectic topology.

Findings

Bound on critical points of Weinstein Morse functions relative to smooth Morse functions

Upper bound on gradient trajectories in Weinstein cobordisms

Vanishing of the Grothendieck group for Weinstein balls

Abstract

We prove that the minimum number of critical points of a Weinstein Morse function on a Weinstein domain of dimension at least six is at most two more than the minimum number of critical points of a smooth Morse function on that domain; if the domain has non-zero middle-dimensional homology, these two numbers agree. There is also an upper bound on the number of gradient trajectories between critical points in smoothly trivial Weinstein cobordisms. As an application, we show that the number of generators for the Grothendieck group of the wrapped Fukaya category is at most the number of generators for singular cohomology and hence vanishes for any Weinstein ball. We also give a topological obstruction to the existence of finite-dimensional representations of the Chekanov-Eliashberg DGA of Legendrian spheres.