  It is well known that in four or more dimensions, there exist exotic
manifolds; manifolds that are homeomorphic but not diffeomorphic to each other.
More precisely, exotic manifolds are the same topological manifold but have
inequivalent differentiable structures. This situation is in contrast to the
uniqueness of the differentiable structure on topological manifolds in one, two
and three dimensions. As exotic manifolds are not diffeomorphic, one can argue
that quantum amplitudes for gravity formulated as functional integrals should
include a sum over not only physically distinct geometries and topologies but
also inequivalent differentiable structures. But can the inclusion of exotic
manifolds in such sums make a significant contribution to these quantum
amplitudes? This paper will demonstrate that it will. Simply connected exotic
Einstein manifolds with positive curvature exist in seven dimensions. Their
metrics are found numerically; they are shown to have volumes of the same order
of magnitude. Their contribution to the semiclassical evaluation of the
partition function for Euclidean quantum gravity in seven dimensions is
evaluated and found to be nontrivial. Consequently, inequivalent differentiable
structures should be included in the formulation of sums over histories for
quantum gravity.
