Principal Investigator: Cameron Gordon
(1) The goal of the project is to investigate several problems in and around 3-dimensional topology. A major focus will be the Cabling Conjecture, which asserts that Dehn surgery on a hyperbolic knot in the 3-sphere always yields a prime 3-manifold. This is part of the program to completely describe all non-hyperbolic Dehn surgeries on hyperbolic knots. Recently, John Luecke and the principal investigator showed that the hyperbolic knots with non-integral toroidal Dehn surgeries are precisely those described by Eudave-Munoz, and some of the techniques developed there may be applicable to the Cabling Conjecture. Another part of the general program that will be addressed is the conjecture that any Seifert fiber space surgery on a hyperbolic knot must be integral. The project will also consider various geometric group theoretic topics that are related to the theory of 3-manifolds, such as surface subgroups of Coxeter and Artin groups, and Gromov's question as to whether or not a 1-ended word hyperbolic group always has a surface subgroup. Another question concerning surfaces and 3-manifolds that will be investigated is the Simple Loop Conjecture, which asserts that if a map from a closed orientable surface to a closed orientable 3-manifold is not injective on fundamental group, then there is an embedded essential loop in the surface whose image is null-homotopic in the 3-manifold.
(2) The project is part of the general goal to understand the structure of 3-dimensional manifolds. These are objects that are locally like ordinary 3-dimensional space, but whose global structure may be quite complicated. Since we live in a 3-manifold, one might say that 3-dimensional topology aims to describe what the mathematical possibilities are for our spatial universe. One important aspect of 3-dimensional topology is the theory of knots - a knot being a closed loop embedded somehow in space. A wide variety of mathematical methods can be applied to the study of knots, leading to new information about 3-manifolds. Recently, deep connections between 3-dimensional topology and quantum physics were discovered in this way. Knot theory is related to the general theory of 3-manifolds through a construction known as Dehn surgery, in which a solid tube around the knot is removed and sewn back in differently. We note that a theorem about Dehn surgery, the Cyclic Surgery Theorem, has been used to determine the topological nature of the action of certain enzymes on strands of DNA. Many aspects of Dehn surgery are now quite well understood. One of the main remaining questions, which is a major focus of the project, is to show that (except in an obvious degenerate situation) the 3-manifold resulting from a Dehn surgery on a knot never decomposes as a "sum" of two simpler manifolds. Another important tool in 3-dimensional topology is the study of (2-dimensional) surfaces in 3-manifolds, and the project will also address various questions in this area. Finally, the project will enable the principal investigator to continue his involvement in the education and training of graduate students in topology at the University of Texas at Austin.
