  The action in general relativity (GR), which is an integral over the manifold
plus an integral over the boundary, is a global object and is only well defined
when the topology is fixed. Therefore, to use the action in GR and in most
approaches to quantum gravity (QG) based on a covariant Lorentzian action,
there needs to exist a prefered (global) timelike vector, and hence a global
topology $R \times S^3$, for it to make sense. This is especially true in the
Hamiltonian formulation of QG. Therefore, in order to do canonical
quantization, we need to know the topology, appropriate boundary conditions and
(in an open manifold) the conditions at infinity, which affects the fundamental
geometrical scalar invariants of the spacetime (and especially those which may
occur in the QG action).
