Topics in low-dimensional topology

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Long plans to work on a variety of projects concerned with the geometric, algebraic and structural aspects of low and higher dimensional hyperbolic, real projective and complex hyperbolic manifolds. The problems form a broad spectrum including an exploration of an intriguing totally new area: the real projective deformations of hyperbolic manifolds. These ideas have connexions with group deformations inside complex hyperbolic isometry groups which it is hoped will have implications for the study of complex hyperbolic manifolds, about which almost nothing is known. Further, the proposer will build upon his recent work on aspects of the questions connected to the existence of surface groups and how one can increase the homology of a hyperbolic manifold, including a new group theoretic perspective on these issues and incorporating new progress in the arithmetic case. Finally, we hope to work on a project which considers pseudo-Anosov maps acting on a punctured torus, which draws together threads coming from elliptic curves, pseudo-Anosov dynamical systems and number theory.A space is called a 3-manifold if it is made of small chunks all of which are ``like'' the ordinary 3-dimensional space that we live in.The recent seminal work of Perelmann has shown that one can get a certain sort of global understanding of such manifolds by proving that they are ``geometric''. Amongst geometrical manifolds, by far the most important are the class called hyperbolic manifolds. This class is ubiquitous in many areas of mathematics, ranging from low-dimensional topology, to dynamical systems, number theory.Indeed, this class is also crucially important in physics. However, even with Perlemann's work, hyperbolic manifolds themselves are still fairly poorly understood, although their importance have made them a magnet for research for well over thirty years. This proposal directs itself towards aspects of the structural study of hyperbolic manifolds and how they can be bent into other types of spaces. The proposer has results which restrict the sorts of manifolds which can arise in certain physical models of the universe and this type of bending is imprtant in physics. He also intends to continue work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.