Twisted generating functions and the nearby Lagrangian conjecture

Abstract

The nearby Lagrangian conjecture predicts that any closed exact Lagrangian submanifold L in the cotangent bundle T * M of a closed manifold M is Hamiltonian isotopic to the zero-section; we refer to such a submanifold L as a nearby Lagrangian. The conjecture is wide open in general and presently known to hold only for M = S 1 (where it is elementary), M = S 2 ([Hin12]) and M = T 2 ([DRGI16]). A first obstruction is that the projection π : L → M may not even be homotopic to a diffeomorphism, or even worse that L may not be diffeomorphic to M. The first and fourth authors have shown in [Kra13] and [Abo12b] that π is at least a homotopy equivalence, a result reproved by the third author in [Gui12]. In [AK18], the first and fourth authors further proved that π is a simple homotopy equivalence. Constraints on the smooth structure of L have also been discovered, e.g. by the first author in [Abo12a] in the case where M is a sphere (see also [EKS15]).We do not pursue this line of research in this paper, but instead focus on the following: if the conjecture holds, then the tangent bundle of L and the cotangent fibres define homotopic sections of the restriction of the Lagrangian Grassmannian bundle of T * M to L. This implies the triviality of the stable Gauss map L → Λ 0 (C ∞ ) (see Definition 2.5, the stable Lagrangian Grassmannian U/O is denoted Λ 0 (C ∞ ) in this paper). Our main result in this regard is the following: Theorem A. Let M be a closed manifold and L a closed exact Lagrangian submanifold of T * M, then the stable Gauss map L → Λ 0 (C ∞ ) vanishes on all homotopy groups.

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

“…Let Σ 0 be an n-dimensional homotopy sphere and let Σ 1 ⊂ T * Σ 0 be a Lagrangian embedding of another homotopy sphere Σ 1 . In [ACGK20] it is proved that the stable Gauss map Σ 1 → U/O is trivial, which is equivalent to the statement that the vertical distribution of T * Σ 0 is stably trivial as a Lagrangian distribution defined along Σ 1 . Therefore, Theorem 1.2 implies the following result.…”

“…This result has the following consequence. In [ACGK20], the triviality of the stable Gauss map Σ 1 → U/O is deduced as a consequence of an existence theorem for generating functions. This theorem states that Σ 1 can be presented as the Cerf diagram of a function f : W → R, where W → Σ 0 is a bundle of tubes in the sense of Waldhausen [W82].…”

“…This theorem states that Σ 1 can be presented as the Cerf diagram of a function f : W → R, where W → Σ 0 is a bundle of tubes in the sense of Waldhausen [W82]. We briefly recall the relevant definitions, and refer the reader to [ACGK20] for futher details.…”

“…Work in progress of the first author with K. Igusa aims to study such Lagrangian homotopy spheres via the the parametrized Morse theory of the generating function f : W → R, thought of as a family of functions on the fibres f x : W x → R, x ∈ Σ 0 . However, theorem [ACGK20] provides no a priori control over the singularities of this family. In particular, there is no guarantee that each f x is Morse or generalized Morse, nor can this be arranged by a generic perturbation.…”

“…Indeed, it is easy to get what we want by 'folding down' the extra dimensions in the factor R 2m , see for example [I88]. We refer the interested reader to [ACGK20], where the appropriate notion of stabilization is discussed in detail. The result is a generating function on the 2m-fold stabilization of W in the sense of Waldhausen's stabilization map T k,N → T k,N +1 [W82], which is still a tube bundle.…”