Essential Surfaces and the Topology of 3-Manifolds

University Of Iowa, Iowa City IA

Investigators

Abstract

DMS-0203394 Ying-Qing Wu The PI of this proposal intends to study the topology of 3-dimensional manifolds via Dehn surgery, essential surfaces and essential laminations. It is a common method to use 2-dimensional surfaces as a tool in studying 3-manifolds. The most useful surfaces in studying 3-manifolds are essential surface, essential laminations, and Heegaard surfaces. The purpose of this project is to study the properties of 3-manifolds using these surfaces and Dehn surgery. There have been great progress in this area in the last few years, for example we now know that an essential surface will remain essential after most Dehn surgeries, and the exact upper bounds on distances between all ten pairs of nonsimple surgery types have been determined. However there are still some very important problems remaining to be solved in this area. An objective of this project is to determine how many surgeries are ``exceptional'' in certain sense, for example, how many will produce so called Seifert manifolds, and how many will kill an essential surface. The PI will also study the problem of how an immersed surface intersects a Heegaard surface. 3-manifolds are objects that are locally like ordinary 3-dimensional spaces, but whose global structure may be quite complicated. Such objects appear naturally from different areas of mathematics as well as physics and other sciences. The ultimate goal in the study of 3-manifold topology is to have a thorough understanding of the structure of 3-manifolds. This has not been achieved yet, but there has been a great progress towards the proof of a conjectural picture described by Thurston more than 20 years ago. The PI proposes to continue studying 3-manifolds via Dehn surgery, which is a method of constructing new 3-dimensional spaces from an existing 3-dimensional space, by removing a neighborhood of a circle and then gluing back in a different way. The achievement of this project will be an important step towards the goal of having a clear understanding of the topological and geometric structures of 3-manifolds.