Nearby Lagrangian conjecture (path-connectedness of L(M))

Prove the nearby Lagrangian conjecture for any closed connected manifold M: the space L(M) of closed connected exact Lagrangian submanifolds in the cotangent bundle T^*M is path-connected, i.e., every such Lagrangian submanifold is Hamiltonian isotopic to the zero-section and therefore diffeomorphic to M.

Background

The paper studies the topology of the space L(M) of closed exact Lagrangian submanifolds in T^*M, motivated by the (strong) nearby Lagrangian conjecture. The classical nearby Lagrangian conjecture predicts that any closed exact Lagrangian in T^*M is Hamiltonian isotopic to the zero section, equivalently that L(M) is path-connected.

The authors develop constraints on the parametrised Whitehead torsion associated to families of nearby Lagrangians, which has implications for the topology of L(M). However, the conjecture itself remains the central motivating open problem in the area.