Contact Manifolds with Flexible Fillings

Abstract

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology; in certain cases, we prove that all flexible fillings are symplectomorphic. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Similar methods are used to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. We also prove relative analogs of our results, which we apply to Lagrangians in cotangent bundles.

Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

“…An important fact is that the property of asymptotically dynamically convex is preserved under subcritical surgeries and flexible surgeries [13,Theorem 3.15,3.17,3.18]. In this note, we will use the following special case.…”

“…One can choose the perturbation carefully such that there is a one-to-one correspondence between Reeb orbits on Y for the contact form λ| Y and pairs of Hamiltonian periodic orbits on [0, ∞) × Y . The differential arises from counting the solutions to the Floer equations; see [4,19,13] for details of the construction. The complex generated by the critical points in W is a subcomplex and the positive symplectic homology SH + * (W ; k) is defined to be the homology of the quotient complex; see [6].…”

“…Remark 2.11. Our grading convention of SH * (W ; k) follows the convention in [3,6,13]. [18,Theorem 6.6] was stated for the symplectic cohomology SH * (W ; k) and it is related to the symplectic homology above by SH * (W ; k) = SH n− * (W ; k).…”

“…Recently, Lazarev [13] showed that all flexible fillings of a contact manifold have the same integral cohomology. He also raised the following question.…”

From symplectic cohomology to Lagrangian enumerative geometry

“…Apart from the foundational material, these tools include confinement lemmas of Cieliebak and Oancea [23], and a lemma about intersection multiplicity with a divisor at constant a periodic orbit by Seidel [58]. Certain ingredients of our proof are also found in other recent papers on related subjects, for example: Diogo [26], Gutt [38], Lazarev [40], Sylvan [63], and the work in preparation of Borman and Sheridan [11].…”