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This can be studied by considering the more general class of quantum field theories which exhibit general covariance. This subject is called topological quantum field theory, or TQFT.
Quantum gravity was originally studied, by Dirac and others, as the problem of quantising general relativity, considered as a dynamical system. This approach has many difficulties, detailed by Isham in
In a TQFT, each manifold gives rise to a quantum propagation law. The mappings of manifolds are symmetries of this theory. In this way, quantum gravity is related to manifold topology.
Without examples, this theory would be pure wishful thinking. There are many examples of TQFTs in 2+1=3 dimensions, some of which are related to quantum gravity in 3 dimensions .
This framework is now leading to suggestions for quantum gravity in 4 dimensions . This is based on the experience with 3 dimensions but is rather more speculative. © 1997 John Barrett