3-dimensional manifolds and related topic

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AbstractAward: DMS-0305846Principal Investigator: Cameron Gordon(1) The goal of the project is to investigate several problems inand around 3-dimensional topology. A major focus will be theCabling Conjecture, which asserts that Dehn surgery on ahyperbolic knot in the 3-sphere always yields a prime3-manifold. This is part of the program to completely describeall non-hyperbolic Dehn surgeries on hyperbolic knots. Recently,John Luecke and the principal investigator showed that thehyperbolic knots with non-integral toroidal Dehn surgeries areprecisely those described by Eudave-Munoz, and some of thetechniques developed there may be applicable to the CablingConjecture. Another part of the general program that will beaddressed is the conjecture that any Seifert fiber space surgeryon a hyperbolic knot must be integral. The project will alsoconsider various geometric group theoretic topics that arerelated to the theory of 3-manifolds, such as surface subgroupsof Coxeter and Artin groups, and Gromov's question as to whetheror not a 1-ended word hyperbolic group always has a surfacesubgroup. Another question concerning surfaces and 3-manifoldsthat will be investigated is the Simple Loop Conjecture, whichasserts that if a map from a closed orientable surface to aclosed orientable 3-manifold is not injective on fundamentalgroup, then there is an embedded essential loop in the surfacewhose image is null-homotopic in the 3-manifold.(2) The project is part of the general goal to understand thestructure of 3-dimensional manifolds. These are objects that arelocally like ordinary 3-dimensional space, but whose globalstructure may be quite complicated. Since we live in a3-manifold, one might say that 3-dimensional topology aims todescribe what the mathematical possibilities are for our spatialuniverse. One important aspect of 3-dimensional topology is thetheory of knots - a knot being a closed loop embedded somehow inspace. A wide variety of mathematical methods can be applied tothe study of knots, leading to new information about3-manifolds. Recently, deep connections between 3-dimensionaltopology and quantum physics were discovered in this way. Knottheory is related to the general theory of 3-manifolds through aconstruction known as Dehn surgery, in which a solid tube aroundthe knot is removed and sewn back in differently. We note that atheorem about Dehn surgery, the Cyclic Surgery Theorem, has beenused to determine the topological nature of the action of certainenzymes on strands of DNA. Many aspects of Dehn surgery are nowquite well understood. One of the main remaining questions, whichis a major focus of the project, is to show that (except in anobvious degenerate situation) the 3-manifold resulting from aDehn surgery on a knot never decomposes as a "sum" of two simplermanifolds. Another important tool in 3-dimensional topology isthe study of (2-dimensional) surfaces in 3-manifolds, and theproject will also address various questions in thisarea. Finally, the project will enable the principal investigatorto continue his involvement in the education and training ofgraduate students in topology at the University of Texas atAustin.