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 nonzero 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 for
Legendrians.
Keywords
Weinstein, Fukaya category, $h$–principle, Grothendieck
group
