Essential laminations and immersed essential surfaces in 3-manifolds

Oklahoma State University, Stillwater OK

Investigators

Abstract

Abstract Award: DMS-0102316 Principal Investigator: Tao Li Essential laminations and immersed essential surfaces are two important objects in 3-manifolds. They generalize embedded incompressible surfaces, and are remarkably useful in obtaining topological and geometric information of 3-manifolds. The main goal of this project is to explore the relationships between the topology of 3-manifolds and these two objects. The tools that the investigator will use include branched surfaces and immersed branched surfaces that have been proved to be extremely useful in the investigator's previous work. The investigator intends to continue his research on essential laminations and immersed surfaces with the following goals. (1) To construct essential laminations in 3-manifolds obtained from Dehn surgery on hyperbolic knots. The techniques to be developed in this research could potentially have great impact on some famous conjectures in knot theory, e.g., Property P for knots and the cabling conjecture. (2) To find an algorithm to decide whether a 3-manifold is a Seifert fiber space. The investigator intends to use immersed branched surfaces and normal surface theory to find a practical algorithm. (3) To show that two homotopy equivalent 3-manifolds, which contain essential laminations, are homeomorphic. (4) To find an algorithm to decide whether a 3-manifold contains an essential lamination. Three-manifolds are objects modeled on the 3-dimensional space that we are living in. These objects can be found in many other sciences, such as physics, biology, and chemistry. A geometric way of studying 3-manifolds, which is extremely fruitful, is to view a 3-manifold as a collection of 3-dimensional pieces glued together along 2-dimensional surfaces. The investigator plans to study the structure of 3-manifolds using such 2-dimensional surfaces, which are called essential laminations and immersed essential surfaces. The research in this project is related to knot theory, which has helped to understand the structure of DNA; it is also related to hyperbolic geometry, which has been used by physicists to understand the universe.