On nonseparating contact hypersurfaces in symplectic 4–manifolds

Abstract

We show that certain classes of contact 3-manifolds do not admit nonseparating contact type embeddings into any closed symplectic 4-manifold, eg this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic 4-manifold. Similarly, there are symplectic 4-manifolds that can admit smoothly embedded nonseparating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.32Q65; 57R17

Contact hypersurfaces in uniruled symplectic manifolds always separate

“…(Actually, the statement of our main theorem for T 2 × S 2 can be proved by more elementary means without mentioning Gromov-Witten invariants; cf. [1,Theorem 1.15].) This implies the well-known fact (see [6]) that the obvious foliation by spheres on S 1 × S 2 cannot be perturbed to a contact structure.…”

“…A construction in dimension 4 was suggested by Etnyre and outlined in [1, Example 1.3]: the idea is to start from a symplectic filling with two boundary components, attach a Weinstein 1-handle to form the boundary connected sum and then attach a symplectic cap to form a closed symplectic manifold, which contains both boundary components of the original symplectic filling as non-separating contact hypersurfaces. At the time [1] was written, examples of symplectic fillings with disconnected boundary were known only up to dimension 6 (due to McDuff [22], Geiges [8,9] and Mitsumatsu [24]), but recently a construction in all dimensions appeared in work of the author with Massot and Niederkrüger [21]. It seems likely that these examples can be combined with the symplectic capping result of Lisca and Matić [17,Theorem 3.2] for Stein fillable contact manifolds to construct examples of non-separating contact hypersurfaces in all dimensions, but we will not pursue this any further here.…”

“…Note that in the above formulation of the Weinstein conjecture for closed contact hypersurfaces, the ambient symplectic manifold need not be closed; for example, every contact manifold is a contact hypersurface in its own (non-compact) symplectization. As was shown in [1], there are many contact manifolds that do not admit any contact-type embeddings into any closed symplectic manifold; as far as the author is aware, all contact manifolds that are currently known to admit such embeddings are also symplectically fillable.…”

“…Under the semipositivity condition, one can show using standard index computations (see [23]) that ev : M A 0,m (M, J) → M m is a pseudocycle of dimension 2(n − 3) + 2c 1 (A) + 2m for generic choices of J ∈ J S 2 , and for such choices, the corresponding Gromov-Witten invariant (without coupling to gravity) can be computed for α 1…”

Periodic symplectic cohomologies

“…We further assume that all Liouville domains in consideration satisfy: (5.5) distinct Reeb orbits x, x have different periods x α = x α. This is a generic condition which can be achieved by perturbing the contact form in the neighborhood of each Reeb orbit as in Theorem 13 of [4]. Rα has an odd number of eigenvalues belonging to (−1, 0), then x is called a negative hyperbolic Reeb orbit.…”