  Using the work of Dwyer, Weiss, and Williams we associate an invariant to any
topologically trivial family of smooth h-cobordisms. This invariant is called
the smooth structure class, and is closely related to the higher
Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure
class in terms of a fiberwise generalized Morse function using fiberwise
Poincar\'e--Hopf theory. This computation gives rise to a duality theorem for
the smooth structure class that generalizes Milnor's duality theorem for the
Whitehead torsion. From this result we deduce a vanishing theorem that implies
the Rigidity Conjecture of Goette and Igusa. This conjecture states that, after
rationalizing, there are no stable exotic smoothings of manifold bundles with
closed even dimensional fibers.
