Generating families and Legendrian contact homology in the standard contact space

Abstract

We show that if a Legendrian knot in a standard contact space R 3 possesses a generating family, then there exists an augmentation of the Chekanov-Eliashberg differential graded algebra so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting, we show that Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation implies normal ruling. ContentsThe second author received support from NSF VIGRE grant no. DMS-0135345 during portions of this work. † The authors became aware of this version of the generating family invariant through a letter from Peter Pushkar to the first author. He considers the homology groups as a possibility for defining a Legendrian homology invariant, but as far as we know has never published anything in this regard. The letter also suggests as a method for computing these groups a complex along the same lines as the one described in Section 5.

Legendrian ambient surgery and Legendrian contact homology

“…For such a Legendrian submanifold it is expected that its Chekanov-Eliashberg algebra has an augmentation whose induced linearisation moreover computes the generating family homology. By the results in [25], this correspondence is known to be true for Legendrian knots inside J 1 R.…”

“…Given that the Legendrian submanifold L ⊂ (J 1 M, λ 0 ) admits a linear-at-infinity generating family as defined in [7], there exists a long exact sequence due to Bourgeois, Sabloff, and Traynor [2] relating the generating family homologies of L S and L. Here L S is obtained by a Legendrian ambient k-surgery on L for any 0 ≤ k ≤ n − 1, under the additional assumption that the generating family on L can be extended over the Lagrangian elementary cobordism V S (see Remark 2.2 for an example when this is not possible). Generating family homology is a Legendrian isotopy invariant for Legendrian submanifolds of J 1 M admitting a generating family; see [38], [40], and [25]. For such a Legendrian submanifold it is expected that its Chekanov-Eliashberg algebra has an augmentation whose induced linearisation moreover computes the generating family homology.…”

Contact Homology, Capacity and Non-Squeezing in \mathbb{R}^{2n}\times S^{1} via Generating Functions

“…They provide a powerful tool in symplectic and contact topology, with important applications also to many of the central problems of these subjects (see for instance Chaperon [4,5], Laudenbach and Sikorav [29], Sikorav [32,33], Givental [22,23], Viterbo [41,39], Traynor [37,38], Théret [34,36,35], Chekanov [6], Eliashberg and Gromov [15], Bhupal [1, 2], Milinković [31], Chekanov and Pushkar [8], Ferrand and Pushkar [18], Jordan and Traynor [28], Colin, Ferrand and Pushkar [12], Chernov and Nemirovski [9,10], Eiseman, Lima, Sabloff and Traynor [13], Fuchs and Rutherford [19]). In particular, Viterbo [41] applied Morse-theoretical methods to the generating function of a Lagrangian submanifold L of the cotangent bundle of a closed manifold B to define invariants c(u, L) ∈ R for any u ∈ H * (B).…”

Knotted Legendrian surfaces with few Reeb chords

“…Remark There is a correspondence between generating families for a Legendrian knot in J 1 .R/ Š R 3 and augmentations for its DGA with coefficients in Z 2 . See eg Fuchs and Rutherford [11]. It is not known whether a similar result holds in higher dimensions.…”