zbMATH — the first resource for mathematics

Morse flow trees and Legendrian contact homology in 1-jet spaces. (English) Zbl 1162.53064
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.: 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.

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
Full Text: DOI arXiv
[1] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 · Zbl 1029.57011
[2] T Ekholm, J Etnyre, L Ng, M Sullivan, The contact homology of conormal lifts of knots and links
[3] T Ekholm, J Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in \(\mathbbR^{2n+1}\), J. Differential Geom. 71 (2005) 85 · Zbl 1098.57013
[4] T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in \(\mathbbR^{2n+1}\), J. Differential Geom. 71 (2005) 177 · Zbl 1103.53048
[5] T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) 453 · Zbl 1076.53099
[6] T Ekholm, J Etnyre, M Sullivan, The Contact Homology of Legendrian Submanifolds in \(P\times\mathbbR\), Trans. Amer. Math. Soc. (to appear) · Zbl 1103.53048
[7] Y Eliashberg, Invariants in contact topology (1998) 327 · Zbl 0913.53010
[8] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560 · Zbl 0989.81114
[9] M Entov, Surgery on Lagrangian and Legendrian singularities, Geom. Funct. Anal. 9 (1999) 298 · Zbl 0994.53030
[10] J B Etnyre, L L Ng, J M Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321 · Zbl 1024.57014
[11] A Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775 · Zbl 0633.53058
[12] A Floer, Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989) 207 · Zbl 0678.58012
[13] A Floer, Monopoles on asymptotically flat manifolds, Progr. Math. 133, Birkhäuser (1995) 3 · Zbl 0836.58010
[14] K Fukaya, Y G Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997) 96 · Zbl 0938.32009
[15] J Milnor, Morse theory, Annals of Mathematics Studies 51, Princeton University Press (1963)
[16] L Ng, Knot and braid invariants from contact homology I, Geom. Topol. 9 (2005) 247 · Zbl 1111.57011
[17] L Ng, Knot and braid invariants from contact homology II, Geom. Topol. 9 (2005) 1603 · Zbl 1112.57001
[18] Y G Oh, Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions, Comm. Pure Appl. Math. 45 (1992) 121 · Zbl 0743.58018
[19] J C Sikorav, Some properties of holomorphic curves in almost complex manifolds, Progr. Math. 117, Birkhäuser (1994) 165
[20] S Smale, On gradient dynamical systems, Ann. of Math. \((2)\) 74 (1961) 199 · Zbl 0136.43702
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.