Topology Seminar (Main Talk): Simplifying Weinstein Morse functions

Seminar | November 8 | 4-5 p.m. | 3 Evans Hall

 Oleg Lazarev, Columbia

 Department of Mathematics

By work of Cieliebak and Eliashberg, any Weinstein structure on Euclidean space that is not symplectomorphic to the standard symplectic structure necessarily has at least three critical points; an infinite collection of such exotic examples were constructed by McLean. Hence Smale's $h$-cobordism theorem fails in the symplectic setting. Nonetheless I will explain why this lower bound on the number of critical points is exact; that is, any Weinstein structure on Euclidean space $\mathbb R^{2n}$ has a compatible Weinstein Morse function with at most three critical points (of index $0$, $n-1$, and $n$). Furthermore, the number of gradient trajectories between the index $n-1$, $n$ critical points can be uniformly bounded. Similarly, any Weinstein structure on the cotangent bundle of the sphere in dimension at least 6 has a compatible Weinstein Morse function with two critical points. As an application, I will construct exotic Weinstein structures on cotangent bundles with many closed exact Lagrangians.

 conway@berkeley.edu