9306240 Raymond Frank Raymond has refined a theory for generalized Seifert fiberings modelled on principal G-bundles to include arbitrary Lie groups G. Applications and examples illustrating how the general theory differs from the case where G is abelian or nilpotent have been developed and will be put into final form. Techniques and results from Seifert fiberings will be used to investigate space form problems for the pseudo-Riemannian manifolds of constant curvature and their frame bundles. Elliptic surfaces whose Euler characteristic are zero support 4-dimensional geometries. A Teichmueller theory for these geometries and the moduli of isometry classes will be constructed and determined. G. Peter Scott plans to work in three main areas: Firstly, he will continue his work on the topological rigidity of 3-manifolds. Secondly, he plans to work in the area of 3-dimensional Poincare duality groups. The long term aim here is to show that any such group comes from a 3-manifold, but he will restrict his attention to those groups which 'ought' to correspond to Seifert fiber spaces or to Haken manifolds. Thirdly, he plans to extend to higher dimensions his work generating the characteristic submanifold of a 3-manifold. Manifolds are natural geometric objects like circles and spheres and doughnuts that can be described locally by the same type of coordinates as a Euclidean space of the same dimension. Their variety and complexity increases rapidly as the dimension increases, but some of the most intractable problems arise already in dimensions three and four. That we live in a world of three dimensions, or four if time is considered, makes this of more than purely academic interest. Cosmologists do not know which 3- or 4-manifold constitutes the physical universe, so studying the possibilities that can occur has ample motivation. These two investigators are making ingenious use of geometry and of algebra to shed light on the question. Their work w ill also enrich the arsenal of tools available to others for studying low-dimensional manifolds. ***
