Problems in Low-Dimensional Topology

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Long plans to work on various old and new projects in geometric,algebraic and structural aspects of low and high dimensional hyperbolicmanifolds. The problems are drawn from a variety of areas: One family ofproblems comes from questions about surface groups, which are of interest not only in low dimensional topology, but also in certain parts of number theory. As an example, we will investigatewhether hyperbolic 3-orbifolds must contain a surface group, orwhether one can increase the homology of a hyperbolic bundle. Anotherfamily of questions come from the study of character varieties. Inhigher dimensions the proposer will consider constructions as well asgeometrical bordism questions coming from physics. Finally, theproposer will work on projects connected to a broad range of questionsarising from the arithmetic of hyperbolic manifolds.A space is called a 3-manifold if it is made of small chunks all ofwhich are ``like'' the ordinary 3-dimensional space that we live in.This proposal directs itself towards aspects of the structural studyof the most important class of 3-manifolds, the so-called hyperbolic3-manifolds. This class is ubiquitous in mathematics and physics, for example although the nature of the universe has yet to be resolved, there is at least one group who have suggested that the universe is the Picard orbifold - a hyperbolic orbifold. The proposer has results which restrict the sorts of manifolds which can arise in certain physical models of the universe. He also intends to contain work on a famous old conjecture about which sorts of two-dimensional objects canlive inside these 3-dimensional objects.