9802558 Wu It is a common practice to use lower dimensional objects as tools for studying manifolds. For example, a manifold containing some embedded essential surfaces, but no essential spheres, is called a Haken manifold. The geometric structure of Haken manifolds is well understood, but unfortunately many manifolds are not Haken. One is thus forced to consider other lower dimensional objects in studying them, such as laminations and immersed surfaces. The objective of this project is to understand the role of immersed essential surfaces in 3-manifolds, particularly the issue of whether such surfaces will lift to embedded surfaces in a finite covering. This would be an important step toward a clear understanding of the topological and geometric structure of 3-manifolds. Manifolds appear naturally in physics and other sciences, since a manifold is an object for which each point is surrounded by a small neighborhood the same as one in the solid ball in a Euclidean space. If the Euclidean space in question is of dimension three, we call such a manifold a 3-manifold. This being the dimension in which we live, it is not surprising that 3-manifolds have attracted a great deal of attention in mathematics and physics. What is perhaps more surprising is the extent to which 3-manifolds have shown themselves resistant to analysis, generally more so than manifolds of dimensions above five, where our 3-dimensional intuitions flounder but other tools come to the rescue. Most of these tools are not available in dimension three, and that is what is at issue here, finding alternatives that do work in the critical dimension three. ***
