Quantum conformal superspace

For a compact connected orientablen-manifoldM, n 3, we study the structure ofclassical superspace,quantum superspace,classical conformal superspace, andquantum conformal superspace. 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 ofM is zero, thenS,S₀,C, andC₀ areilhorbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS₀ and quantum conformal superspaceC₀ are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractibleilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS₀ andC₀. Our results are applicable to questions involving the problem of thereduction 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 spaceC₀ will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.