QUANTUM CONFORMAL SUPERSPACE

For a compact connected orientable n-manifold M, n greater than or equ al to 3, we study the structure of classical superspace S = M/D, quant um 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 st ructure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity t o a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degr ee 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 tru e, we show that quantum superspace S-o and quantum conformal superspac e C-o are in fact ILH-manifolds. If, moreover, M is a Haken manifold, then quantum superspace and quantum conformal superspace are contracti ble ILH-manifolds. In this case, there are no Gribov ambiguities for t he configuration spaces S-o and C-o. Our results are applicable to que stions involving the problem of the reduction of Einstein's vacuum equ ations and to problems involving quantization of the gravitational fie ld. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equati ons to an unconstrained Hamiltonian system on a reduced phase space. F or the problem of quantum gravity, the space C-o will play a natural r ole in any quantization procedure based on the use of conformal method s and the reduced Hamiltonian formulation.