Maximal contact and symplectic structures

Abstract

We study the relationship on Weinstein domains given by Weinstein cobordism. Our main result is that any finite collection of high‐dimensional Weinstein domains with the same topology is Weinstein subdomains of a ‘maximal’ Weinstein domain also with the same topology. As applications, we construct many new exotic Weinstein structures, for example, exotic cotangent bundles containing many closed regular Lagrangians that are formally Lagrangian isotopic but not Hamiltonian isotopic and a new exotic Weinstein structure on Euclidean space. A novel feature of our construction of exotic structures is the use of results from symplectic flexibility. We also describe a similar construction in the contact setting which we use to produce ‘maximal’ contact structures and extend several existing results in low‐dimensional contact geometry to high dimensions. We prove that all contact manifolds have symplectic caps, introduce a general procedure for producing contact manifolds with many Weinstein fillings, and give a new proof of the existence of codimension two contact embeddings.

Planarity in higher-dimensional contact manifolds

“…That is, Stein fillable contact manifolds admit symplectic caps. For general higher-dimensional contact manifolds, it was not known until very recently that they admit (strong) symplectic caps [6,23]. This fact can be used to recover Theorem A (with the word "weakly" replaced by "strongly") from Theorem B, but our proof is independent of the existence of caps.…”

Contact surgery and symplectic caps

“…Remark While completing a draft of this note, Oleg Lazarev sent a draft of his paper [21] in which he establishes Theorem 1.1 for almost‐Weinstein fillable contact manifolds , using isotropic surgery of arbitrary index and coisotropic surgery of index .…”

“…Remark While completing a draft of this note the authors learned that Oleg Lazarev had also obtained Theorem 1.6, see [21]. In addition, he showed that the partial order on contact manifolds coming from Weinstein cobordisms in dimensions above 3 has upper bounds for finite sets.…”

A landscape of contact manifolds via rational SFT

“…In the context of Weinstein cobordism, there are geometric constructions of a contact manifold Y that is Weinstein cobordant both from Y 1 and Y 2 given that there are Weinstein cobordism from Y to Y 1 , Y 2 in dimension ≥ 5 [49]. However it is not clear if such construction would yield something larger in Con ≤ or Con ≤,W .…”

“…On the other hand, when we consider strong cobordisms, there are a lot more morphisms. In particular, if we include ∅ into the discussion, the existence of symplectic cap [25,49] implies that anything with a strong filling is equivalent to ∅ in Con ≤,S . Even if we throw out ∅ and even restrict to the case of connected strong cobordisms, the existence of strong cobordism is much less rigid compared to the the counterparts for exact or Weinstein cobordisms by [74].…”