  Two singularity theorems can be proven if one attempts to let a Lorentzian
cobordism interpolate between two topologically distinct manifolds. On the
other hand, Cartier and DeWitt-Morette have given a rigorous definition for
quantum field theories (qfts) by means of path integrals. This article uses
their results to study whether qfts can be made compatible with topology
changes. We show that path integrals over metrics need a finite norm for the
latter and for degenerate metrics, this problem can sometimes be resolved with
tetrads. We prove that already in the neighborhood of some cuspidal
singularities, difficulties can arise to define certain qfts. On the other
hand, we show that simple qfts can be defined around conical singularities that
result from a topology change in a simple setup. We argue that the ground state
of many theories of quantum gravity will imply a small cosmological constant
and, during the expansion of the universe, will cause frequent topology
changes. Unfortunately, it is difficult to describe the transition amplitudes
consistently due to the aforementioned problems. We argue that one needs to
describe qfts by stochastic differential equations, and in the case of gravity,
by Regge calculus in order to resolve this problem.
