  We show how the path integral for gravity and matter on a piecewise flat
spacetime can be used to define the physical quantum gravity states and the
related transition amplitudes. The physical states are given by the path
integrals for open manifolds from a certain topological class, while the
corresponding transition amplitudes are obtained by gluing two such open
manifolds into a closed one and calculating the corresponding path integral. We
also solve the problem of how to associate a quantum gravity state to an
effective action. This is done by using the path integral for a closed manifold
from the transition-amplitude topological class so that the state which
corresponds to the effective action can be choosen to be the Hartle-Hawking or
the Vilenkin wavefunction for the vacuum manifold component of the
transition-amplitude manifold.
