Families of Legendrian submanifolds via generating families

Abstract

We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded in the standard contact R 2n`1 (for n ą 1) that are contractible in the smooth, but not Legendrian, categories.

Lagrangian cobordisms via generating families: Construction and geography

“…In this section, we illustrate the possibilities of the construction in two families of examples. As noted in the introduction, deeper applications of these constructions appear in [6,8,28,38].…”

A Legendrian Turaev torsion via generating families

“…Under this isomorphism, Alexander duality for generating family homology corresponds to Sabloff duality for linearized contact homology [Sab06]. Generating family homology can also be used to construct invariants of families of Legendrians, as in the work of Sabloff and Sullivan [SS16]. Finally, we mention the connection with rulings [FI04], [Sab05].…”

The minimal length of a Lagrangian cobordism between Legendrians

“…(b) Via an additional surgery, a connect sum of two copies of this sphere produces a sphere that has a front projection that is invariant under a 180 • rotation of the x-coordinates. This rotation produces a non-trivial loop of Legendrian spheres [38].…”

“…isotopies The non-trivial loop γ that leads to long cobordisms comes from a loop of Legendrian trefoil knots constructed by Kálmán [32]. The non-trivial loop γ that leads to short cobordisms comes from a loop of Legendrian spheres in R 2n+1 constructed by the first author and Sullivan [38].…”