  The gauge symmetry of classical general relativity under space-time
diffeomorphisms implies that any path integral quantization which can be
interpreted as a sum over space-time geometries, gives rise to a formal
invariant of smooth manifolds. This is an opportunity to review results on the
classification of smooth, piecewise-linear and topological manifolds. It turns
out that differential topology distinguishes the space-time dimension d=3+1
from any other lower or higher dimension and relates the sought-after path
integral quantization of general relativity in d=3+1 with an open problem in
topology, namely to construct non-trivial invariants of smooth manifolds using
their piecewise-linear structure. In any dimension d<=5+1, the classification
results provide us with triangulations of space-time which are not merely
approximations nor introduce any physical cut-off, but which rather capture the
full information about smooth manifolds up to diffeomorphism. Conditions on
refinements of these triangulations reveal what replaces block-spin
renormalization group transformations in theories with dynamical geometry. The
classification results finally suggest that it is space-time dimension rather
than absence of gravitons that renders pure gravity in d=2+1 a `topological'
theory.
