Simplifying Weinstein Morse functions

Lazarev, Oleg

doi:10.2140/gt.2020.24.2603

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Simplifying Weinstein Morse functions

Oleg Lazarev

Geometry & Topology 24 (2020) 2603–2646

DOI: 10.2140/gt.2020.24.2603

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

References

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

Volume 24, issue 5 (2020)