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On nonseparating contact hypersurfaces in symplectic 4-manifolds. (English) Zbl 1207.32019

A closed hypersurface \(M\) in a symplectic manifold \((W,\omega)\) is of contact type if it is transverse to a Liouville vector field \(Y\). If \(M\) is such a hypersurface, then \(i_Y\omega\) gives a contact form on \(M\).
The paper discusses conditions for contact 3-manifolds to occur as non-separating hypersurfaces in closed \(4\)-manifolds. The main theorem asserts that obstructions to non-separating embeddings include Giroux torsion, planar open book decompositions, or the existence of a symplectic cap with an embedded sphere of non-negative self-intersection. The theorem’s conclusion remains true even if its conditions are only satisfied after performing contact connected sums and/or contact \(-1\) surgeries on \(M\). For \(M\) with Giroux torsion this implies that no contact embedding into a 4-manifold exists at all, separating or not. The symplectic cap obstruction is in place, among other examples, for all embedded contact 3-submanifolds of the standard \(\mathbb R^4\).
From another angle, if the ambient 4-manifold \(W\) contains an embedded sphere of non-negative self-intersection, e.g., if it is rational or ruled, then it has no contact non-separating hypersurfaces. In particular, some obvious non-separating hypersurfaces in \(\Sigma\times S^2\), like \(l\times S^2\) with \(l\) being a non-separating closed curve, can not be of contact type. For embedded spheres of positive self-intersection, the result follows from McDuff’s characterization of such \(W\) as blow-ups of \(S^2\times S^2\) or \(\mathbb{CP}^2\) making them simply connected. Based on their results, the authors conjecture that all contact 3-manifolds admitting non-separating embeddings also admit separating ones, and that even separating embeddings are impossible for \(W\) rational or ruled.
The proofs are based on a technique developed recently by Wendl which uses foliations of symplectic 4-manifolds by punctured pseudoholomorphic curves. In the case of symplectizations, such foliations are typically lifted from planar open book decompositions of the underlying contact 3-manifolds. Results of this paper require a generalization to partially planar open books, i.e., contact fiber sums along bindings of open books at least one of which is planar. For example, the standard torus \(T^3\) is partially planar but not planar, and other such contact structures on \(\Sigma\times S^1\) can be readily constructed. The assumptions of the main theorem permit the construction of a compact moduli of pseudoholomorphic curves that foliate a non-compact symplectic manifold, giving a contradiction.

MSC:

32Q65 Pseudoholomorphic curves
57R17 Symplectic and contact topology in high or arbitrary dimension
53D05 Symplectic manifolds (general theory)
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