Mathematical Sciences: Essential Laminations and the Topology of 3-Manifolds

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The main goal of this project is to explore the interrelationships between the topology of a 3-manifold and the types of laminations and foliations it can support. The investigator will look into the nature of group actions on the 2- sphere and their extensions into the 3-ball. The main questions he plans to consider are the following. (1) If M is irreducible, N is laminated, and M and N are homotopy equivalent, are they homeomorphic? (2) If M is laminated, then are homotopic homeomorphisms isotopic? (3) What manifolds have essential laminations? (4) If G is a 2-sphere convergence group which extends to a properly discontinuous action on the 3-ball, is that extension unique up to conjugation? The study of 3-dimensional manifolds is aided by the fact that our experience of living in such a space endows us with a strong intuition for what can and cannot take place. Studying the topology of higher dimensional manifolds is necessarily dependent upon the algebraic machinery available to assist in it, but in 3 dimensions we also have at our disposal a direct avenue of perception. It is therefore surprising to learn that some questions that have natural generalizations to higher dimensions have been answered already for these higher cases, while the 3- dimensional case that inspired them remains obdurately unassailable. There is a classical conjecture of Poincare about spheres that is the most notorious example of this phenomenon. What appears to be at work in thus violating our natural over- estimate of the advantage that geometric intuition should confer in 3 dimensions? It is at least partly a naive faith in intuition and mistrust of computation, but it is also the fact that the known methods of computation perform best with the luxury of excess dimensions in which to maneuver. They are somehow cramped in the presence of only three dimensions. In this setting one can see that the investigator's exploitation of laminations as a tool particularly adapted to the 3-dimensional environment and our intuition about it constitutes a welcome and enormously promising development.