A note on the front spinning construction

Abstract

ABSTRACT. In this paper we introduce a notion of front S m -spinning for Legendrian submanifolds of R 2n+1 . It generalizes the notion of front S 1 -spinning which was invented by Ekholm, Etnyre and Sullivan. We use it to prove that there are infinitely many pairs of exact Lagrangian cobordant and not pairwise Legendrian isotopic Legendrian S 1 × S i1 × · · · × S i k which have the same classical invariants if one of i j 's is odd.

On Legendrian cobordisms and generating functions

“…We use a well-known construction in Legendrian geometry called front spinning. It consists of including the wave front of a Legendrian submanifold in a bigger space, and then rotating it using the additionnal coordinates in order to create a bigger wave front (see Ekholm-Etnyre-Sullivan [8] and Golovko [13]).…”

Families of Legendrian submanifolds via generating families

“…This link, which is isotopic to the Hopf link, has a generating family f : RˆR N Ñ R with the the top strand of the top component generated by critical points of index r`N and the bottom strand of the bottom component generated by critical points of index N´1. Spin the front about its central axis into R n`1 as in [10] to get two Legendrian spheres. Then perform a 0-surgery along the horizontal dotted 1-disk in Figure 1 to get a connected Legendrian sphereΛ n,r .…”

“…In the next few subsections, we bring the front spinning constructions of [6,10], their adaptation to generating families [2], and their generalization to twist spinning [2] into the families context.…”

“…If m " 1, we must explicitly assume that the lifting procedure yields a loop, not just a path, of generating families. As a common generalization of [2] and [10], and in parallel to [7] for m " 1, we define a generating family for the twist-spun Legendrian pn`mq-submanifold Λ α by:…”

On Lagrangian embeddings of closed nonorientable 3–manifolds

“…,ỹ(p),z(p)) to satisfyx > 0. Applying the front S 1 -spinning construction [13] toh 0 , we obtain the Lagrangian filling F :…”