Curves and 3-Manifolds

University Of California-Davis, Davis CA

Investigators

Abstract

We describe three problems in this proposal, all in the general area of knot theory and 3-manifolds. A Heegaard splitting of a 3-manifold is a decomposition of the 3-manifold into simple pieces, called handlebodies. This decomposition is an effective way to study many of the interesting open questions about 3-manifolds, but there are some startling gaps in our basic information about Heegaard splittings. We know that given two Heegaard splittings of the same 3-manifold, one can move from one to the other by a series of moves. The first problem in this proposal asks how "far apart" two different Heegaard splittings of a 3-manifold can be under these moves. The second problem is to explore a particular aspect of the Poincare Conjecture prompted by Dunwoody's recent attempt and suggested by earlier work of Freedman and Yau. We will look carefully at whether a plan analogous to what Dunwoody proposed can be made to work on a real 3-ball, beginning with an even simpler version of the question on a triangulated 2-sphere. The final problem is a generalization of recent work of the proposer on immersed curves in the plane to immersed 2-spheres in 3-space. We will look at the set of singularities (including, for example, curves of self-intersection and points of zero curvature) and try to derive relationship between numbers and types of singularities, similar to the kinds of results known for immersed curves in the plane or in projective 2-space. We also describe how these and other related problems have been used by the proposer to stimulate interest in the field and in mathematics in general at levels ranging from high school students, including undergraduates and graduate students, through postdocs. Low-dimensional topology is the study of properties of spaces in dimensions two, three, and four. It is a field that used to lie squarely in the realm of "pure" mathematics, and research in the field was pursued largely for its intricacy and beauty. As we continue to understand the universe, from the shape of space itself to the knotting of strands of DNA, the deep connections between this abstract area and the real world are increasingly apparent. This proposal aims to explore some of the fundamental questions in 3-dimensional spaces and knot theory, including understanding the relationships between different decompositions of 3-dimensional spaces and quantifying the complexity of 2-dimensional spheres immersed in a standard 3-dimensional universe.