Invariants in Low Dimensional Topology

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DMS-0305012James ConantThe principal investigator, a low dimensional topologist, is utilizing atool he developed with Peter Teichner (called ``grope cobordism") to findtopological applications and interpretations for an important class ofinvariants called Vassiliev invariants. In particular he and Teichner arepursuing the question of whether Vassiliev invariants detect knottedness.The principal investigator is also looking for more powerful versions ofthe theory of Vassiliev invariants, hopefully giving rise to fourdimensional (concordance) information. Some of this is joint with JacobMostovoy and Ted Stanford. Working jointly with Matt Horak and KarenVogtmann, the principal investigator is studying the mapping class groupof surfaces via a tool developed by Harer, Penner, and elucidated byKontsevich (graph homology). Graph homology is very similar to the algebrathat appears in the context of Vassiliev invariants. The principalinvestigator is also studying the group of outer automorphisms of a freegroup via graph homology. Finally, the principal investigator, workingjointly with Ryan Budney, Kevin Scannell, and Dev Sinha, is pursuing aprogram for grounding Vassiliev invariants in classical homotopy theoryand differential topology. This program has already borne fruit in termsof geometric interpretations of the simplest Vassiliev invariant.The yoga behind the theory of topological invariants is that they provideuseful, computable topological information. When the number of dimensionsis large, complete answers to topological questions can often be foundusing algebraic invariants. In low dimensions, such as 3 and 4, thesituation is more complicated. The invariants still exist, but they aremuch weaker, and usually do not give complete answers. On the other hand,there are lots of invariants in low dimensions which don't have highdimensional analogues, and which are poorly understood topologically. Forexample, there is a polynomial you can associate to any knotted loop (suchas a piece of DNA or a singularity in spacetime) which will not changeeven if the loop is pushed and pulled into a new shape. This polynomial,called the Jones polynomial, has a definition which eludes any topologicalinterpretation and whose topological applications have been modest. Theprincipal investigator is working to understand precisely the connectionof this polynomial (and other similar objects) with topology, and thiswill lead to more effective methods for answering questions in lowdimensional topology and beyond.