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Simplifying Weinstein Morse functions

Oleg Lazarev

Geometry & Topology 24 (2020) 2603–2646

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.

Weinstein, Fukaya category, $h$–principle, Grothendieck group
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53D37, 53D40, 57R80
Received: 8 May 2019
Revised: 7 November 2019
Accepted: 9 December 2019
Published: 29 December 2020
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Leonid Polterovich
Oleg Lazarev
Department of Mathematics
Columbia University
New York, NY
United States