Wendl, Chris A hierarchy of local symplectic filling obstructions for contact 3-manifolds. (English) Zbl 1279.57019 Duke Math. J. 162, No. 12, 2197-2283 (2013). A symplectic manifold \((W,\omega)\) is said to have convex boundary if on a neighborhood of \(\partial\,W\) there exists a vector field \(Y\) that points transversely outward at \(\partial\,W\) and whose flow is a symplectic dilation, that is, \({\mathcal L}_Y\omega=\omega\). The cooriented hyperplane field \(\xi=\ker(\iota_Y\omega_{|T\partial W})\subset T\partial W\) satisfies a certain maximal nonintegrability condition which makes it a contact structure, and up to isotopy, it depends only on the symplectic structure of \((W,\omega)\) near \(\partial\,W\), not on the choice of vector field \(Y\). It is interesting to ask which isomorphism classes of contact manifolds do not arise as boundaries of compact symplectic manifolds, that is, which ones are not symplectically fillable. There are examples of obstructions to symplectic filling that depend on the global properties of the manifolds involved. In contrast, one can also consider filling obstructions which are local, that is, whether some contact subdomains can never exist in the convex boundary of a compact symplectic manifold. There are examples of symplectic filling obstructions that contact-type boundaries of symplectic 4-manifolds can never contain an overtwisted disk, and those furnished by the so-called Giroux torsion domain.In this paper, the author introduces a geometric formalism and expands the known repertoire of local filling obstructions. In particular, it is demonstrated that the presented examples occupy the first two levels in an infinite hierarchy: for each integer \(k\geq 0\), a special class of compact contact 3-manifolds, possibly with boundary, which are called planar \(k\)-torsion domains, are defined. The definition of planar torsion combines the fundamental contact topological notion of a supporting open book decomposition with a simple topological operation known as the contact fiber sum along codimension 2 contact submanifolds. For a closed contact 3-manifold \((M,\xi)\) with nondegenerate contact form \(\lambda\) and generic compatible complex structure \(J:\xi\to\xi\) a chain complex is defined by so-called orbit sets \(\gamma=\left((\gamma_1,m_1),\dots,(\gamma_n,m_n)\right)\), where \(\gamma_i\) are distinct simply covered periodic Reeb orbits and \(m_i\) are positive integers, called multiplicities. A differential operator is then defined by counting a certain class of embedded rigid \(J\)-holomorphic curves in the symplectization of \((M,\xi)\), which can be viewed as cobordisms between orbit sets. The homology of the resulting chain complex is called the embedded contact homology \(ECH_*(M,\lambda,J)\). The empty orbit set, which is always a cycle in the homology, defines a distinguished class \(c(\xi)\) that is called the ECH-contact invariant. The main results of the paper state that if \((M,\xi)\) is a closed contact 3-manifold with planar torsion of any order, then it does not admit a contact-type embedding into any closed symplectic 4-manifold and its ECH-contact invariant \(c(\xi)\) vanishes. If \((M,\xi)\) is a closed contact 3-manifold with fully separating planar torsion, then its twisted ECH-contact invariant \(c(\xi)\) vanishes. It is also shown that a closed contact 3-manifold has planar \(0\)-torsion if and only if it is overtwisted and every closed contact manifold with Giroux torsion also has planar \(1\)-torsion. If \((M,\xi)\) is a contact 3-manifold supported by an open book \(\pi:M\setminus B\to S^1\), then any planar torsion domain in \((M,\xi)\) must intersect the binding \(B\). Further properties state that if \((M,\xi)\) is a closed contact 3-manifold that contains a partially planar domain and admits a contact-type embedding \(\iota:(M,\xi)\to(W,\omega)\) into some closed symplectic 4-manifold \((W,\omega)\), then \(\iota\) separates \(W\), and for all integers \(d\geq 1\), the image of \(U^d:ECH_*(M,\xi)\to ECH_*(M,\xi)\) contains \(c(\xi)\). As the technical basis of these results, the author establishes existence, uniqueness, and compactness theorems for certain classes of \(J\)-holomorphic curves in blown-up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. Finally, the author presents some open questions and discusses the recent progress in the subject. Reviewer: Andrew Bucki (Edmond) Cited in 1 ReviewCited in 15 Documents MSC: 57R17 Symplectic and contact topology in high or arbitrary dimension 53D10 Contact manifolds (general theory) 32Q65 Pseudoholomorphic curves 53D42 Symplectic field theory; contact homology 57M50 General geometric structures on low-dimensional manifolds Keywords:overtwistedness; Giroux torsion; contact manifold; infinite hierarchy of local filling obstruction; planar torsion; contact fiber sum; embedded contact homology × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] C. Abbas, Holomorphic open book decompositions , Duke Math. J. 158 (2011), 29-82. · Zbl 1230.53077 · doi:10.1215/00127094-1276301 [2] C. Abbas, K. Cieliebak, and H. Hofer, The Weinstein conjecture for planar contact structures in dimension three , Comment. Math. Helv. 80 (2005), 771-793. · Zbl 1098.53063 · doi:10.4171/CMH/34 [3] P. Albers, B. Bramham, and C. 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