On the parametrised Whitehead torsion of families of nearby Lagrangian submanifolds

TL;DR

This paper investigates the parametrised Whitehead torsion of families of Lagrangian submanifolds in cotangent bundles, showing it factors through simpler maps and has implications for the Lagrangian monodromy problem.

Contribution

It extends previous work by analyzing the torsion's properties and triviality on certain homotopy groups using twisted generating functions.

Findings

Whitehead torsion is trivial on pi_0 and pi_1

Torsion's image is divisible by the Euler characteristic

Provides implications for high-dimensional torus Lagrangian monodromy

Abstract

Motivated by the strong nearby Lagrangian conjecture, we constrain the parametrised Whitehead torsion of a family of closed exact Lagrangian submanifolds in a cotangent bundle. We prove the parametrised Whitehead torsion admits a factorisation through simpler maps, in particular implying it is trivial on $\pi_0$, $\pi_1$, and that its image is divisible by the Euler characteristic. We provide concrete implications for the Lagrangian monodromy question in the case of a high dimensional torus. This generalises earlier work of Abouzaid and Kragh \cite{AbKr} on the $\pi_0$ version, using different methods. Our main tool is the theory of twisted generating functions, building on \cite{ACGK}.