Rational symplectic field theory over \(\mathbb Z_{2}\) for exact Lagrangian cobordisms.

*(English)*Zbl 1154.57020Let \(X\) be a symplectic \(2n\)-manifold with a cylindrical ends and \(L\) a Lagrangian \(n\)-submanifold such that outside a compact subset, the pair \((X, L)\) is symplectomorphic to the disjoint union of \((Y^+\times\mathbb R_+, \Lambda^+\times\mathbb R_+)\) and \((Y^-\times\mathbb R_-, \Lambda^-\times\mathbb R_-)\), where \(\Lambda^{\pm}\subset Y^{\pm}\) is a Legendrian \((n-1)\)-submanifold. Let \((\overline X, \overline L)\) denote the compact part of \((X, L)\). The author calls \((X, L)\) an exact cobordism if the symplectic form \(\omega\) on \(X\) satisfies \(\omega=d\beta\) for some \(1\)-form \(\beta\) and \(\beta| _L=df\) for some function \(f\). The author considers a formal disk in \((X, L)\), that is a homotopy class of maps of the 2-disk \(D\) with some marked disjoint closed subintervals in \(\partial D\) into \(\overline X\), where the marked intervals are required to map in orientation preserving (reversing) manner to Reeb chords of \(\partial\overline L\) in the \((+\infty)\)-boundary (in the \((-\infty)\)-boundary) and where the remaining parts of the boundary \(\partial D\) map to \(\overline L\). The action of a Reeb chord is the integral of the contact form \(\beta\) in the \((\pm\infty)\)-boundary along it and the \((+)\)-action of the formal disk is the sum of the actions of its Reeb chords in the \((+\infty)\)-boundary. The author introduces the notion of an admissible formal disk in \((X, L)\) so that if a formal disk determined by a \(J\)-holomorphic disk in \(X\) with boundary on \(L\) is admissible then bubbling is impossible for topological reasons.

Let \(\mathbf V(X, L)\) denote the graded vector space over \(\mathbb Z_2\) consisting of all formal sums of admissible disks of \((X, L)\) which contain only a finite number of summands of the \((+)\)-action below any given number and with grading such that the degree of a formal disk is the formal dimension of the moduli space of \(J\)-holomorphic disks homotopic to the formal disk. The filtration \(0\subset F^k\mathbf V(X, L)\subset \cdots\subset F^2\mathbf V(X, L) \subset F^1\mathbf V(X, L)=\mathbf V(X, L)\) with respect to the number \(k\) of pieces of \(L\) is defined where the filtration level is determined by the number of Reeb chords of a formal disk in the \((+\infty)\)-boundary of \((X, L)\). The author defines a differential \(d: \mathbf V(X, L)\to\mathbf V(X, L)\) which counts all admissibe formal disks which are built from the following disks by gluing at Reeb chords: one admissible \(J\)-holomorphic disk in the symplectization of the \((\pm\infty)\)-boundary of degree 1, one copy of the admissible formal disks and any number of admissible \(J\)-holomorphic disk in \((X, L)\) of degree 0. Thus \(d\) increases grading by 1, respects the filtration and is \((+)\)-action non-decreasing.

For \(\alpha>0\), the finite \((+)\)-action subspace \(\mathbf V_{[\alpha]}(X, L) \subset\mathbf V(X, L)\) is defined by the subspace of formal sums of admissible disks of \((+)\)-action at most \(\alpha\). The differential \(d\) induces filtration preserving differentials \(d_{\alpha}\) in \(\mathbf V_{[\alpha]}(X, L)\) and the natural projection map \(\pi^{\alpha}_{\beta}: \mathbf V_{[\alpha]}(X, L)\to\mathbf V_{[\beta]}(X, L), \alpha>\beta\) is a filtration preserving chain map. The author obtains the rational admissible SFT spectral sequence \(\{E^{p,q}_{r, [\alpha]}(X, L)\}^k_{r=1}\) as the spectral sequence induced by \(d_{\alpha}\). By restricting the contact form of the contact manifolds at \((\pm\infty)\)-boundary of cobordism, the spectral sequence is invariant under compactly supported deformations (Theorem 1.1). If \(\Lambda\subset Y\) is a Legendrian submanifold of a contact manifold \(Y\) and if \(\Lambda\) is subdivided into pieces \(\Lambda=\Lambda_1\cup\cdots\cup \Lambda_k\subset Y\), then the spectral sequence \(\{E^{p,q}_r(Y, L)\}^k_{r=1}=\underset\alpha{\underleftarrow {\lim}} \{E^{p,q}_{r, [\alpha]}(Y\times\mathbb R, \Lambda\times\mathbb R)\}^k_{r=1}\) gives Legendrian isotopy invariants of \(\Lambda\) (Theorem 1.2).

Let \(\mathbf V(X, L)\) denote the graded vector space over \(\mathbb Z_2\) consisting of all formal sums of admissible disks of \((X, L)\) which contain only a finite number of summands of the \((+)\)-action below any given number and with grading such that the degree of a formal disk is the formal dimension of the moduli space of \(J\)-holomorphic disks homotopic to the formal disk. The filtration \(0\subset F^k\mathbf V(X, L)\subset \cdots\subset F^2\mathbf V(X, L) \subset F^1\mathbf V(X, L)=\mathbf V(X, L)\) with respect to the number \(k\) of pieces of \(L\) is defined where the filtration level is determined by the number of Reeb chords of a formal disk in the \((+\infty)\)-boundary of \((X, L)\). The author defines a differential \(d: \mathbf V(X, L)\to\mathbf V(X, L)\) which counts all admissibe formal disks which are built from the following disks by gluing at Reeb chords: one admissible \(J\)-holomorphic disk in the symplectization of the \((\pm\infty)\)-boundary of degree 1, one copy of the admissible formal disks and any number of admissible \(J\)-holomorphic disk in \((X, L)\) of degree 0. Thus \(d\) increases grading by 1, respects the filtration and is \((+)\)-action non-decreasing.

For \(\alpha>0\), the finite \((+)\)-action subspace \(\mathbf V_{[\alpha]}(X, L) \subset\mathbf V(X, L)\) is defined by the subspace of formal sums of admissible disks of \((+)\)-action at most \(\alpha\). The differential \(d\) induces filtration preserving differentials \(d_{\alpha}\) in \(\mathbf V_{[\alpha]}(X, L)\) and the natural projection map \(\pi^{\alpha}_{\beta}: \mathbf V_{[\alpha]}(X, L)\to\mathbf V_{[\beta]}(X, L), \alpha>\beta\) is a filtration preserving chain map. The author obtains the rational admissible SFT spectral sequence \(\{E^{p,q}_{r, [\alpha]}(X, L)\}^k_{r=1}\) as the spectral sequence induced by \(d_{\alpha}\). By restricting the contact form of the contact manifolds at \((\pm\infty)\)-boundary of cobordism, the spectral sequence is invariant under compactly supported deformations (Theorem 1.1). If \(\Lambda\subset Y\) is a Legendrian submanifold of a contact manifold \(Y\) and if \(\Lambda\) is subdivided into pieces \(\Lambda=\Lambda_1\cup\cdots\cup \Lambda_k\subset Y\), then the spectral sequence \(\{E^{p,q}_r(Y, L)\}^k_{r=1}=\underset\alpha{\underleftarrow {\lim}} \{E^{p,q}_{r, [\alpha]}(Y\times\mathbb R, \Lambda\times\mathbb R)\}^k_{r=1}\) gives Legendrian isotopy invariants of \(\Lambda\) (Theorem 1.2).

Reviewer: Haruo S. Suzuki (Sapporo)

##### MSC:

57R17 | Symplectic and contact topology in high or arbitrary dimension |

57R58 | Floer homology |

53D40 | Symplectic aspects of Floer homology and cohomology |

53D12 | Lagrangian submanifolds; Maslov index |

##### Keywords:

holomorphic curve; Lagrangian submanifold; Legendrian submanifold; symplectic cobordism; symplectic field theory##### References:

[1] | Bourgeois, F.: A Morse-Bott approach to contact homology. PhD thesis, Stanford University (2002) · Zbl 1046.57017 |

[2] | Bourgeois, F., Ekholm, T., Eliashberg, Y.: A surgery exact sequence for linearized contact homology. In preparation |

[3] | Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799-888 (2003) · Zbl 1131.53312 |

[4] | Bourgeois, F., Mohnke, K.: Coherent orientations in symplectic field theory. Math. Z. 248, 123-146 (2004) · Zbl 1060.53080 |

[5] | Chekanov, Y.: Differential algebra of Legendrian links. Invent. Math. 150, 441-483 (2002) · Zbl 1029.57011 |

[6] | Cornea, O., Lalonde, F.: Cluster homology: an overview of the construction and results. Electron. Res. Announc. Amer. Math. Soc. 12, 1-12 (2006) · Zbl 1113.53052 |

[7] | Ekholm, T.: Morse flow trees and Legendrian contact homology in 1-jet spaces. Geom. Topol. 11, 1083-1224 (2007) · Zbl 1162.53064 |

[8] | Ekholm, T.: Legendrian 2n-spheres distinguished by rational SFT. In preparation |

[9] | Ekholm, T., Etnyre, J., Ng, L., Sullivan, M.: Knot contact homology. In preparation · Zbl 1267.53095 |

[10] | Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in 2n+1 R . J. Differential Geom. 71, 85-128 (2005) · Zbl 1098.57013 |

[11] | Ekholm, T., Etnyre, J., Sullivan, M.: The contact homology of Legendrian submanifolds in 2n+1 R . J. Differential Geom. 71, 177-305 (2005) · Zbl 1103.53048 |

[12] | Ekholm, T., Etnyre, J., Sullivan, M.: Orientations in Legendrian contact homology and exact Lagrangian immersions. Int. J. Math. 16, 453-532 (2005) · Zbl 1076.53099 |

[13] | Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in P \times R. Trans. Amer. Math. Soc. 359, 3301-3335 (2007) · Zbl 1119.53051 |

[14] | Eliashberg, Y.: Invariants in contact topology. In: Proc. Int. Congress Math., Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 327-338 · Zbl 0913.53010 |

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.