Mathematical Sciences: Topics in Low-Dimensional Topology

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The study of 3-manifolds is often enriched by imposing extra structure on the manifold: geometric structure such as a Riemannian metric of negative curvature, or topological-dynamical structure such as a flow or lamination. The two investigators will continue research into 3-manifold supporting laminations and pseudo-Anosov flows. They will also undertake new research into the structure of 3-manifolds and more general spaces which are not negatively curved. In a joint project, Mosher and Oertel will explore a new technique for analyzing such spaces. Any such space can be made to support a certain kind of 2-dimensional measured lamination. The properties of this lamination will be studied and used in an attempt to shed light on the general structure of these spaces. In a separate project, Mosher will continue a study of Thurston's homology norm for a 3-manifold, and techniques for computing this norm using pseudo-Anosov flows. He will also pursue a computer study of ends of hyperbolic 3- manifolds. In still another separate project, Oertel will continue research into a class of 3-manifolds called laminated manifolds. The first goal is to extend methods and theorems from the well-known class of Haken manifolds to the larger class of laminated manifolds. The second goal is to show that, in some sense, "most" 3-manifolds are laminated. It is a surprising fact that, although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3- dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3-dimensional spheres, where precisely the original 3- dimensional case is the only one still open. The investigators are pursuing a variety of questions about 3-dimensional manifolds, some with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.