Topological Aspects of Classical and Quantum Gravity.

The question of which 3-manifolds occur as hypersurfaces of physically reasonable spacetimes has been a problem of interest for several years. R. Schoen and S. T. Yau proved that for asymptotically flat 3-manifolds with positive energy and no apparent horizons the manifolds are the connected sum of manifolds with finite fundamental group and handles. It will be shown that there are no restrictions on which closed 3-manifolds occur as hypersurfaces of vacuum spacetimes and all occur in the asymptotically flat case if the horizon condition is removed. An implication of this work is the surprising result that spacetimes with maximal hypersurfaces are the exception rather than the rule: all 3-manifolds occur as spacelike hypersurface of asymptotically flat spacetimes with positive energy but only the ones listed by Schoen-Yau can be maximal hypersurfaces. In canonical quantum gravity, the group of disconnected components of the diffeomorphisms of a fixed hypersurface form a symmetry group of the quantum states associated with the hypersurface. In the present work techniques are developed to calculate these groups. These techniques are applicable to any 3-manifold, but will only be applied to a special class of 3-manifolds. The reason for only looking at this special class of 3-manifolds is they are best understood and the most basic 3-manifolds. The 3 -manifolds studied here are constructed from a 3-sphere by identifying points via isometries of the sphere. For this class of 3-manifolds the symmetry groups are comprised of SU(2) coverings of SO(3) crystallographic groups, the product of these with a cyclic group, cyclic groups and the production of two cyclic groups. The structure of these is very different compared to the cyclic groups encountered in Yang-Mills theory. Using these groups, a complete classification of which 3-manifolds have quantum states with half integral angular momentum is given. As a by-product of this work, several interesting mathematical results are found. First, the symmetry groups themselves are of interest to mathematicians. Second, it is shown that for spherical spaces most of them never admit orientation reversing diffeomorphisms. Finally, the first example of a diffeomorphism of 3-manifold which is homotopic but not isotopic to the identity is given.