9704811 Brittenham Essential laminations are one of a large array of objects that are used to probe the structure of 3-manifolds. They were developed as a result of an attempt to generalize simultaneously two such classical objects; the incompressible surface and the taut foliation. This project involves studying the structure of essential laminations and how they sit inside of a 3-manifold, to extract information about the `laminar' 3-manifold. The main goals that the investigator will pursue are to show that every irreducible 3-manifold homotopy-equivalent to a laminar manifold is homeomorphic to it, and to show that laminar manifolds are finitely covered by Haken manifolds. He will also study taut foliations from the laminar point of view, using this new point of view to study these older objects. The goal is to find examples of manifolds that admit essential laminations but which do not contain either of the two `parent' objects. An associated goal is to find a hyperbolic 3-manifold which does not admit an essential lamination. A 3-dimensional manifold is an object which looks like our ordinary 3-dimensional space, if you don't look too far away from where you are standing. Such objects can behave very differently over large distances, though; walking in a straight line, for example, can bring you right back to where you started from (much like walking on the surface of the spherical Earth). 3-manifold topology attempts to describe this global structure. In doing so, ideas are drawn from, and applied to, many other branches of science; they have been applied to chemistry, where they have helped to unlock some of the secrets of DNA recombination, and they have been both drawn from and applied to physics, in work aimed at understanding the basic building blocks of the universe. One technique that has proved fruitful in studying 3-manifolds is to think of the 3-manifold as a collection of surfaces (called a foliation) stacked together (like the pages of a book); then one can use what is known about surfaces to explore the structure of the 3-manifold. The investigator plans to continue work aimed at developing this approach. ***
