Lagrangian cobordisms via generating families: Construction and geography

Abstract

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobor-dism developed by Biran and Cornea. This yields a non-degenerate Lagrangian "spectral metric" which bounds the Lagrangian "cobordism metric" (recently introduced by Cornea and Shelukhin) from below. It also yields a new numerical Lagrangian cobordism invariant as well as new ways of computing certain asymp-totic Lagrangian spectral invariants explicitly.

Legendrian ambient surgery and Legendrian contact homology

“…Remark 1.8. 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].…”

“…The neighbourhood theorem [26, Theorem 6.2.2] for isotropic submanifolds gives neighbourhoods and a map φ as above satisfying (1) and (2). We may furthermore assume that (3) holds infinitesimally, i.e.…”

“…In Section 4 we provide a generalisation of cusp-connect sum to handle the case of a general k-surgery by a construction that we call Legendrian ambient surgery. A related construction has also appeared in [2].…”

Lagrangian fillings of Legendrian 4-plat knots

“…With the notion of Lagrangian cobordism in hand, we define the central object for this paper. [3]; similar constructions appear in [11,14]. See Figure 7 for an example of a Lagrangian filling of a Legendrian 5 2 knot using Theorem 3.3.…”

On Legendrian cobordisms and generating functions

“…For the last twenty years, the most studied notion has been the one of Lagrangian cobordism between Legendrian submanifolds. To the search for obstructions to the existence of such cobordisms initiated by Baptiste Chantraine in [5], Lisa Traynor and her collaborators successfully added the ingredient of gf's and came out with the notion of gf-compatible Lagrangian cobordisms [3], [16]. For instance, Sabloff and Traynor in [16] found out that there is no gfcompatible Lagrangian cobordisms between the two Chekanov-Eliashberg knots.…”