Open books and configurations of symplectic surfaces.

*(English)*Zbl 1035.57015Let \((X, \omega)\) 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.

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.

Reviewer: Haruo S. Suzuki (Sapporo)

##### 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) |

**OpenURL**

##### References:

[1] | S Akbulut, B Ozbagci, On the topology of compact Stein surfaces, Int. Math. Res. Not. (2002) 769 · Zbl 1007.57023 |

[2] | Y Eliashberg, Filling by holomorphic discs and its applications, London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press (1990) 45 · Zbl 0731.53036 |

[3] | D T Gay, Explicit concave fillings of contact three-manifolds, Math. Proc. Cambridge Philos. Soc. 133 (2002) 431 · Zbl 1012.57039 |

[4] | D T Gay, Symplectic 2-handles and transverse links, Trans. Amer. Math. Soc. 354 (2002) 1027 · Zbl 0992.57026 |

[5] | E Giroux, in preparation |

[6] | M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 · Zbl 0592.53025 |

[7] | A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of \(B^4\), Invent. Math. 143 (2001) 325 · Zbl 0983.32027 |

[8] | D McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991) 651 · Zbl 0719.53015 |

[9] | D McDuff, D Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1998) · Zbl 0844.58029 |

[10] | M Symington, Symplectic rational blowdowns, J. Differential Geom. 50 (1998) 505 · Zbl 0935.57035 |

[11] | M Symington, Generalized symplectic rational blowdowns, Algebr. Geom. Topol. 1 (2001) 503 · Zbl 0991.57027 |

[12] | W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345 · Zbl 0312.53028 |

[13] | I Torisu, Convex contact structures and fibered links in 3-manifolds, Internat. Math. Res. Notices (2000) 441 · Zbl 0978.53133 |

[14] | B Wajnryb, An elementary approach to the mapping class group of a surface, Geom. Topol. 3 (1999) 405 · Zbl 0947.57015 |

[15] | A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241 · Zbl 0737.57012 |

[16] | D T Gay, Open books and configurations of symplectic surfaces, Algebr. Geom. Topol. 3 (2003) 569 · Zbl 1035.57015 |

[17] | D T Gay, R Kirby, Constructing symplectic forms on 4-manifolds which vanish on circles, Geom. Topol. 8 (2004) 743 · Zbl 1054.57027 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.