#### Volume 24, issue 5 (2020)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Simplifying Weinstein Morse functions

### Oleg Lazarev

Geometry & Topology 24 (2020) 2603–2646
##### 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 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
##### Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53D37, 53D40, 57R80
##### Publication
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
##### Authors
 Oleg Lazarev Department of Mathematics Columbia University New York, NY United States