  The Chern-Weil topological theory is applied to a classical formulation of
general relativity in four-dimensional spacetime. Einstein--Hilbert
gravitational action is shown to be invariant with respect to a novel
translation (co-translation) operator up to the total derivative; thus, a
topological invariant of a second Chern class exists owing to Chern-Weil
theory. Using topological insight, fundamental forms can be introduced as a
principal bundle of the spacetime manifold. Canonical quantization of general
relativity is performed in a Heisenberg picture using the Nakanishi-Kugo-Ojima
formalism in which a complete set of quantum Lagrangian and BRST
transformations including auxiliary and ghost fields is provided in a
self-consistent manner. An appropriate Hilbert space and physical states are
introduced into the theory, and the positivity of these physical states and the
unitarity of the transition matrix are ensured according to the Kugo-Ojima
theorem. The nonrenormalizability of quantum gravity is reconsidered under the
formulation proposed herein.
