Studies in low-dimensional topology

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Many of the basic problems in 3-dimensional topology can be analysed in terms of the Dehn filling operation, a method for transforming one 3-dimensional space into another. One of the main goals of my proposal is to contribute to our understanding of this operation and to apply this work to study open problems such as the exceptional filling conjecture and the generalised knot complement conjecture. Topologists often associate algebraic objects and their algebraic properties to topological spaces and their topological properties. My second goal is to investigate relations between the three quite different measures of largeness for a 3-dimensional space W. First, the existence of a co-oriented taut foliation in W, a topological condition concerning partitions of W into surfaces. Second, the left-orderability of the group of W, a group theoretic condition concerning the compatibility of multiplication in the group with a linear order on it. Third, the property that W not be a Heegaard-Floer L-space, an analytic condition. Though there is no compelling heuristic which explains why these conditions should be closely connected, there is no known W for which they differ and many infinite families for which they are known to be the same. Advances in our understanding of the topology of 3-dimensional spaces have led us to the point where we can fruitfully investigate relations between them. I will study the qualitative behavior of infinite families of projections, or non-zero degree maps, of one 3-manifold onto another, with the goal of showing that under suitable conditions, such families arise as by-products of a fixed projection. Another important relation between spaces is that of commensurability. Two spaces are called commensurable if they can be finitely unfolded to yield the same space. Another goal of my proposal is to show that if a hyperbolic knot space is commensurable to another knot space, then the topology of the associated knots is very special.