Surfaces in low-dimensional topology

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

The investigator proposes to study the topology of 3-dimensional manifolds, using 2-dimensional laminations and foliations as the basic tools. The main goal is to show how these objects can be used to attack several long-standing conjectures in low-dimensional topology. In particular, he proposes to use constructions of essential laminations, due to the investigator and others, to classify the exceptional Dehn surgeries on alternating knots. He plans to use laminations to show that some large classes of 3-manifolds satisfy the Virtual Haken Conjecture. He will also continue work aimed at providing a better understanding of the structure of sutured manifold decompositions, a fundamental technique in building and using taut foliations, by developing examples of hyperbolic knots with large depth. We live in and move through a 3-dimensional universe, which we perceive as extending infinitely in all directions. But recent observations hint at the possibility that space may instead wrap around itself, so that we live in a finite universe. It is this very idea of a space wrapping around itself that lies at the heart of geometric topology, whose goal is to understand the global structure of objects which locally look like ordinary Euclidean space. The investigator plans to study several questions aimed at uncovering patterns in the global structure of 3-dimensional spaces, by using 2-dimensional surfaces to study the spaces in which they can be found. The main idea is to use surfaces as a way of cutting the space into pieces; by studying the resulting pieces, we can gain insight into the large-scale structure of the 3-dimensional space.