Quantum conformal superspace

For a compact connected orientable n-manifold M, n greater than or equal to 3, we study the structure of classical superspace S = M/D, quantum superspace S-o = M/D-o, classical conformal superspace C = (M/P)/D, and quantum conformal superspace C-o = (M/P)/D-o. The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry of M is zero, then S, S-o, C, and C-o are ILH-orbifolds. The case of most importance for general relativity is dimension n = 3. In this case, assuming that the extended Poincare conjecture is true, we show that quantum superspace S-o and quantum conformal superspace C-o are in fact ILH-manifolds. If, moreover, M is a Haken manifold, then quantum superspace and quantum conformal superspace are contractible ILH-manifolds. In this case, there are no Gribov ambiguities for the configuration spaces S-o and C-o. Our results are applicable to questions involving the problem of the reduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the space C-o will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.