Studies in Low-Dimensional Topology

数学家的一项主要任务是充分了解我们所居住的空间的类型，以便能够列出它们所有可能的形状，并准确地描述这些三维形状的结构。为了做到这一点，数学家将几何空间和几何问题转化为可以进行计算的代数语言。例如，我们可以将一个称为“群”的代数对象与每个三维空间相关联，并且在许多情况下，具有相同群的空间必然是相同的。因此，理解不同可能的三维空间是由理解可能出现的不同可能群的代数问题决定的。我的研究计划的主要目标之一是研究将三维空间的某些性质转化为代数的可能性。所讨论的几何性质是将一个空间切割成不相交的表面的能力，这些表面局部地拼凑在一起，就像一副纸牌。并不是所有的三维空间都能像这样被分割，预计只有那些群可以以代数相干的方式排序的空间才能这样做。这就是我打算调查的。这种空间分裂的存在似乎也等同于反映更高维几何的某种解析条件，这相当令人惊讶，因为没有令人信服的启发式方法来解释为什么这三个条件应该联系在一起。另一方面，没有已知的空间它们是不同的，而许多无穷无尽的家庭是已知的相同的。我这部分研究计划的另一个目标是调查解析条件是否与几何条件相同。*我的研究建议的第二部分涉及Dehn填充操作，这是一种将一个三维空间转换到另一个三维空间的方法。三维空间的许多基本问题都可以用Dehn填充来分析，我的主要目标之一是帮助我们理解这个运算，然后应用这项工作来研究三维空间的不同形状。*我们对三维空间形式的认识的进步使我们能够卓有成效地研究它们之间的关系。在我的提案的第三部分，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。这将导致更好地理解这些空间是如何关联的。最后，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。我的提议的最终目标是了解给定类型的多少不同空间可以是可公度的，并准确地描述那些可与其他空间公度的空间。

A major task for mathematicians is to understand the types of spaces that we live in well enough to be able to list all their possible shapes and accurately describe each of these 3-dimensional shape's structure. To do this, mathematicians convert geometric spaces and geometric problems into the language of algebra, where calculations can be made. For instance, to each 3-dimensional space we can associate an algebraic object called a "group", and in many cases spaces with the same group are necessarily the same. Thus understanding the different possible 3-dimensional spaces is determined by the algebraic problem of understanding the different possible groups which can arise. One of the principal goals of my research proposal is to study the possibility of translating certain properties of 3-dimensional spaces into algebra. The geometric property in question is the ability to cut a space into disjoint surfaces which piece together locally like a deck of cards. Not all 3-dimensional spaces can be cut up like this and it is expected that only those whose groups can be ordered in algebraically coherent way do. This is what I intend to investigate. It also seems that the existence of such a splitting of the space is equivalent to a certain analytic condition reflecting higher dimensional geometry, and this is quite surprising as there is no compelling heuristic which explains why these three conditions should be connected. On the other hand, there is no known space for which they differ and many infinite families for which they are known to be the same. Another goal of this part of my research program will be to investigate whether the analytic condition is the same as the geometric one. ******A second part of my research proposal concerns the Dehn filling operation, a method for transforming one 3-dimensional space into another. Many of the basic problems of 3-dimensional spaces can be analysed in terms of Dehn filling, and one of my main goals is to contribute to our understanding of this operation and then to apply this work to study the different shapes of 3-dimensional spaces. ******Advances in our knowledge of the form of 3-dimensional spaces have led us to the point where we can fruitfully investigate relations between them. In a third part of my proposal 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. This will lead to a better understanding of how such spaces relate. Finally 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. A final goal of my proposal is to understand how many different spaces of a given type can be commensurable and precisely describe those which are commensurable to some other space.