Studies in low-dimensional topology

三维拓扑学中的许多基本问题都可以通过Dehn填充操作来分析，Dehn填充操作是一种将一个三维空间转换到另一个空间的方法。我的建议的主要目的之一是帮助我们理解这一运算，并将这一工作应用于研究公开问题，如例外填充猜想和广义纽结补猜想。 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中三种完全不同的大小度量之间的关系。第一，W中是否存在共同定向的拉紧叶层，这是一种关于W划分为曲面的拓扑条件。第二，W的群的左序性，这是一个关于群中乘法与其上的线性序相容的群论条件。第三，W不是Heegaard-Floer L空间的性质，这是一个解析条件。虽然没有令人信服的启发式方法来解释为什么这些条件应该紧密地联系在一起，但没有已知的W是它们不同的，也没有许多无穷无尽的家庭知道它们是相同的。 在我们对三维空间拓扑学的理解上的进步使我们能够卓有成效地研究它们之间的关系。我将研究从一个三维流形到另一个流形的无限族的投影族或非零度映射的定性行为，目的是证明在适当的条件下，这种族作为固定投影的副产品出现。空间之间的另一个重要关系是可公度性。如果两个空间可以有限地展开以产生相同的空间，则称它们为可公度空间。我的建议的另一个目的是证明如果一个双曲纽结空间与另一个纽结空间是可公度的，那么相关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.