Studies in Low-Dimensional Topology

数学家的一项主要任务是了解我们生活的空间类型，以便能够列出所有可能的形状，并准确描述每种三维形状的结构。为了做到这一点，数学家将几何空间和几何问题转换成代数语言，在代数语言中可以进行计算。例如，对于每个三维空间，我们可以将一个称为“群”的代数对象联系起来，并且在许多情况下，具有同一群的空间必然相同。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空间切割成不相交的表面的能力，这些表面在局部拼接在一起，就像一副纸牌。并不是所有的三维空间都可以这样分割，而且只有那些群可以以代数相干的方式排序的空间才可以这样分割。这就是我要调查的。似乎这种空间分裂的存在等同于反映高维几何的某种分析条件，这是相当令人惊讶的，因为没有令人信服的启发式来解释为什么这三个条件应该联系起来。另一方面，没有已知的空间表明它们是不同的，而且有许多无限的家族表明它们是相同的。我这部分研究计划的另一个目标是研究解析条件是否与几何条件相同。我的研究计划的第二部分涉及Dehn填充操作，这是一种将一个三维空间转换为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操作，然后应用这个工作来研究三维空间的不同形状。我们对三维空间形式的知识的进步使我们能够卓有成效地研究它们之间的关系。在我的建议的第三部分，我将研究无限族的投影，或非零度映射，一个3-流形到另一个的定性行为，目的是表明，在适当的条件下，这些族作为一个固定投影的副产品出现。这将有助于更好地理解这些空间之间的关系。最后，空间之间的另一个重要关系是可通约性。如果两个空间可以被有限地展开以产生相同的空间，则它们被称为可通约的。我建议的最终目标是了解给定类型中有多少不同的空间是可通约的，并精确描述那些与其他空间可通约的空间。

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.