Studies in low-dimensional topology

三维拓扑学中的许多基本问题都可以用德恩填充操作来分析，德恩填充操作是一种将一个三维空间转换为另一个三维空间的方法。我的建议的主要目标之一是有助于我们理解这个操作，并应用这项工作来研究开放的问题，如特殊填充猜想和广义结补猜想。 拓扑学家经常将代数对象及其代数性质与拓扑空间及其拓扑性质联系起来。我的第二个目标是研究三维空间W的三种完全不同的大度量之间的关系。首先，存在一个共同定向绷紧叶理在W，拓扑条件的分区W成表面。第二，W的群的左序性，一个关于群中乘法与其上线性序相容的群论条件;第三，W不是Heegaard-Floer L-空间的性质，一个解析条件。虽然没有令人信服的启发式来解释为什么这些条件应该是密切相关的，但没有已知的W，它们不同，并且有许多无限族，它们已知是相同的。 我们对三维空间拓扑的理解的进步使我们能够富有成效地研究它们之间的关系。我将研究一个3流形到另一个3流形的无限投影族或非零度映射的定性行为，目的是表明在适当的条件下，这些族是作为固定投影的副产品出现的。空间之间的另一个重要关系是可互换性。如果两个空间可以反向展开以产生相同的空间，则它们被称为可反向展开的空间。我的建议的另一个目标是表明，如果一个双曲纽结空间是可扩展到另一个纽结空间，那么相关的纽结的拓扑是非常特殊的。

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.