Parametrised Whitehead Torsion of Families of Nearby Lagrangians

The parametrised Whitehead torsion is an invariant of families of manifolds, and can be viewed as a map to an algebraic K-theory space. A strong version of the nearby Lagrangian conjecture says that when applied to families of closed exact Lagrangians in a cotangent bundle, this invariant vanishes.

Abouzaid and Kragh showed that in this case, this map lands in the trivial path component of the target, i.e. is trivial on π0

. Using generating functions, we find strong constraints on what this map does to higher homotopy groups. I'll illustrate this with some concrete consequences for the symplectic mapping class group of T∗Tn

relative to the 0-section.

This is based on joint work-in-progress with Sylvain Courte.