Over the years a nice conjectural picture has emerged of the structure of closed 3-manifolds. It has been shaped by the geometrization and virtual bundle conjectures of Thurston and the compactification conjectures of Simon and of Marden. These imply certain consequences for covering spaces of such 3-manifolds. Closed, orientable, irreducible 3-manifolds with infinite fundamental group ought to have universal covering space homeomorphic to Euclidean 3-space. Professor Myers will continue his work on this problem by ruling out classes of possible counterexamples through an analysis of group actions on Whitehead manifolds; he will use a new result which promises to extend his work on Whitehead manifolds which contain non-trivial planes to those which do not. More generally, covering spaces with finitely generated fundamental group ought to be homeomorphic to interiors of compact 3-manifolds. He will continue his work on this problem and on the closely related problem of showing that the same conclusion holds for hyperbolic 3-manifolds with finitely generated fundamental group. Freedman has reduced the latter problem to a purely topological problem which seems accessible. Hyperbolic 3-manifolds of finite volume ought to be finitely covered by surface bundles over the circle, although very few non-trivial examples of this phenomenon are known. Myers will attempt to construct more such examples and will explore the extent to which the desired conclusion is implied by the failure of the finitely generated intersection property in the fundamental group. 3-manifolds with cubings of negative curvature ought to be hyperbolic; Myers will investigate a method for trying to build hyperbolic structures, trying to determine both its scope and limitations. Finally, he will attempt to construct small knots in non-Haken 3-manifolds.
Manifolds are spaces which generalize the familiar curves and surfaces that live in ordinary 3-dimensional space. An n-manifold locally looks like n-dimensional space; thus a curve is a 1-manifold, and a surface is a 2-manifold. 3-manifolds are difficult to visualize but can be described mathematically. Such spaces arise as models of the universe in general relativity; in this role they come equipped with geometries, i.e. ways of measuring distances, angles, and curvature. Conjectures of Thurston, Simon, and Marden suggest that the members of a large class of 3-manifolds have particularly nice geometries and can be constructed from the members of a small set of standard 3-manifolds by identifying their points in certain ways. Professor Myers will attempt to verify various important special cases of this conjectural picture.
