Morse flow trees and Legendrian contact homology in 1-jet spaces.

*(English)*Zbl 1162.53064Let \(M\) be a Riemannian \(n\)-manifold and let \(J^1(M):=T^*M \times\mathbb{R}\) be its 1-jet space. \(J^1(M)\) is endowed with the standard contact structure \(\xi\) defined as the kernel of the standard 1-form on \(T^*M\times\mathbb{R}\). An \(n\)-dimensional submanifold \(L\) of \(J^1(M)\) is called “Legendrian” if it is everywhere tangent to \(\xi\). In the paper under review, the author studies Legendrian submanifolds \(L\subset J^1 (M)\) with a view toward their Legendrian contact homology. In symplectic field theory [cf.: Y. Eliashberg, A. Givental and H. Hofer, in: Geom. Funct. Analysis, Basel: Birkhäuser, 560–673 (2000; Zbl 0989.81114)], Legendrian contact homology is a useful framework for finding isotopy invariants of Legendrian submanifolds of contact manifolds by appropriately counting rigid (pseudo-)holomorphic disks, where the analytical foundations of this toolkit were established in a series of foregoing papers by T. Ekholm, J. Etnyre and M. Sullivan, Trans. Am. Math. Soc. 359, No. 7, 3301–3335 (2007; Zbl 1119.53051)].

In the present paper, these investigations are continued by constructing an explicit correspondence between certain rigid flow trees in \(M\) associated to the Legendrian submanifold \(L\subset J^1(M)\) and boundary punctured pseudo-holomorphic disks in the cotangent bundle \(T^*M\) of particular boundary and asymptotic behaviour with respect to the projection of \(L\). This result makes it possible to reduce the computation of Legendrian contact homology in \(J^1(M)\) to a finite-dimensional problem in Morse theory, which in turn has powerful applications to establishing new invariants of knots in 3-space [cf.: L. Ng, Geom. Topol. 9, 247–297, 1603–1637 (2005; Zbl 1111.57011 and Zbl 1112.57001)]. As for the basic objects linked by the author’s main theorem, a flow tree in \(M\) determined by the Legendrian submanifold \(L\subset J^1(M)\) is a continuous map of a combinatorial tree \(\Gamma\) to \(M\) such that the restriction of the map to any edge of \(\Gamma\) parametrizes a part of a gradient flow line of some local function difference. A tree is called rigid if its formal dimension is zero and if it is transversally cut out in a certain sense. These fundamental objects are thoroughly discussed in the first two sections of the paper, whereas the subsequent five sections are devoted to the proof of the above-mentioned correspondence theorem, its consequences, and its applications. The paper comes with a very lucid and detailed introduction explaining the main results and the general strategy of this important, equally detailed and comprehensive work.

In the present paper, these investigations are continued by constructing an explicit correspondence between certain rigid flow trees in \(M\) associated to the Legendrian submanifold \(L\subset J^1(M)\) and boundary punctured pseudo-holomorphic disks in the cotangent bundle \(T^*M\) of particular boundary and asymptotic behaviour with respect to the projection of \(L\). This result makes it possible to reduce the computation of Legendrian contact homology in \(J^1(M)\) to a finite-dimensional problem in Morse theory, which in turn has powerful applications to establishing new invariants of knots in 3-space [cf.: L. Ng, Geom. Topol. 9, 247–297, 1603–1637 (2005; Zbl 1111.57011 and Zbl 1112.57001)]. As for the basic objects linked by the author’s main theorem, a flow tree in \(M\) determined by the Legendrian submanifold \(L\subset J^1(M)\) is a continuous map of a combinatorial tree \(\Gamma\) to \(M\) such that the restriction of the map to any edge of \(\Gamma\) parametrizes a part of a gradient flow line of some local function difference. A tree is called rigid if its formal dimension is zero and if it is transversally cut out in a certain sense. These fundamental objects are thoroughly discussed in the first two sections of the paper, whereas the subsequent five sections are devoted to the proof of the above-mentioned correspondence theorem, its consequences, and its applications. The paper comes with a very lucid and detailed introduction explaining the main results and the general strategy of this important, equally detailed and comprehensive work.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

53D40 | Symplectic aspects of Floer homology and cohomology |

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

53D35 | Global theory of symplectic and contact manifolds |

53D10 | Contact manifolds (general theory) |

57R58 | Floer homology |

##### Keywords:

symplectic and contact homology; Legrendrian submanifolds; jet spaces; Legendrian contact homology; Floer homology symplectic aspects; Morse theory; symplectic field theory##### References:

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