Monodromy of loops of exact Lagrangians: pseudo-isotopy and isotopy to the identity

Determine, for a closed connected manifold M, whether the monodromy diffeomorphism φ ∈ Diff(M) arising from any loop (L_t)_{t∈[0,1]} of closed exact Lagrangian submanifolds in T^*M based at the zero-section (L_0=L_1=M) is pseudo-isotopic to the identity and, moreover, whether φ is isotopic to the identity.

Background

Given a loop of nearby Lagrangians based at the zero-section, there is an associated monodromy in the mapping class group of M. The projection L_t→M being a homotopy equivalence implies the monodromy is homotopic to the identity, but the finer questions of pseudo-isotopy and isotopy remain unresolved in general.

The authors obtain partial results for the case M=T^n (high-dimensional tori), but they explicitly note that they are unable to establish the pseudo-isotopy or isotopy conclusions in general.