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Open books and configurations of symplectic surfaces. (English) Zbl 1035.57015
Let $$(X, \omega)$$ be a symplectic $$4$$-manifold such that $$\omega\wedge\omega >0$$. A symplectic configuration in $$(X, \omega)$$ is a union $$C=\Sigma_1\cup \dots\cup\Sigma_n$$ of embedded closed surfaces such that all intersections between surfaces are $$\omega$$-orthogonal. To this, the author associates a labelled graph $$G(C)$$ with no edges from a vertex to itself, where each vertex corresponds to a $$\Sigma_i$$ and is labelled by a triple of numbers, consisting of the genus $$g_i$$ of $$\Sigma_i$$, the self-intersection $$\Sigma_i\cdot\Sigma_i$$ and the area $$a_i=\int_{\Sigma_i}\omega$$, and where each edge represents a point of intersection. Because $$\omega$$-orthogonal intersections are necessarily positive, $$G(C)$$ completely determines the topology of regular neighborhood of $$C$$, that is, the result of plumbing disk bundles over surfaces according to $$G(C)$$. For any vertex $$v_i$$ let $$d_i$$ denote the number of edges connected to $$v_i$$. If $$m_i+d_i>0$$ for every $$v_i$$, $$G(C)$$ is called positive.
The boundary of $$(X, \omega)$$ is called concave (resp. convex) if there exists a symplectic dilation $$V$$ defined on a neighborhood of $$\partial X$$ pointing in (resp. out) along $$\partial X$$, which induces a negative (resp. positive) contact structure $$\xi=ker\iota_V\omega\mid_{\partial X}$$. The main theorem states that positive symplectic configurations have neighborhoods with concave boundaries and explicitly describes the contact structures on such boundaries in terms of open book decompositions. The proof uses the author’s results from [Math. Proc. Camb. Philos. Soc. 133, 431–441 (2002; Zbl 1012.57039) and Trans. Am. Math. Soc. 354, 1027–1047 (2002; Zbl 0992.57026)] on contact surgeries and symplectic handlebodies alongside the construction by A. Weinstein [Hokkaido Math. J. 20, 241–251 (1991; Zbl 0737.57012)]. As applications, the author gets criteria for some contact structure supported by an open book to be strongly symplectically fillable.

##### MSC:
 57R17 Symplectic and contact topology in high or arbitrary dimension 57N10 Topology of general $$3$$-manifolds (MSC2010) 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
##### Keywords:
symplectic configuration; contact; concave; open book; plumbing; fillable
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