Simplifying Weinstein Morse functions

Lazarev, Oleg¹

¹ Department of Mathematics, Columbia University, New York, NY, United States

Geometry & topology, Tome 24 (2020) no. 5, pp. 2603-2646.

Résumé

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.

Détail

DOI : 10.2140/gt.2020.24.2603

Classification : 57R17, 53D37, 53D40, 57R80
Keywords: Weinstein, Fukaya category, $h$–principle, Grothendieck group

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