Morse flow trees and Legendrian contact homology in 1-jet spaces
Let \(M\) 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.: \textit{Y. Eliashberg, A. Givental} and \textit{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 \textit{T. Ekholm, J. Etnyre} and \textit{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.: \textit{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.
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