Low dimensional topology, ordered groups and actions on 1-manifolds

低维拓扑、有序群和 1-流形上的动作

• 批准号: RGPIN-2020-05343
• 负责人: Clay, Adam
• 金额: $ 1.97万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758496
• 关键词: Low; dimensional; topology; ordered; groups

Groups are mathematical objects developed from a study of symmetries and rigid motions, though in modern research there are many places where groups spring up as a sophisticated way of encoding certain kinds of data. It is natural that one of the challenges, then, is to learn how to extract the data encoded by a group from a study of its algebraic properties. One method of tackling this challenge is to show that your group of interest is isomorphic to (the same as) a subgroup of another group that is better understood. I employ this method in my research, where the common theme is to try to realize a given group as a subgroup of the group of order-preserving homeomorphisms of a low-dimensional space (that is, as a collection of deformations of a low-dimensional object) such as the real line or the circle. More generally, it is sometimes useful to realize the group as a collection of order-preserving functions from an arbitrary ordered set to itself. The goal of this program is then to study the different ways that a group can be realized as subgroups of deformations of the real line, the circle or an ordered set, and to extract algebraic information about a given group from these structures. Of particular importance is the case when a group arises from the study of a 3-dimensional space via a construction known as the "fundamental group". In this case, the goal becomes to extract from the fundamental group information about the topology of the underlying space; and in this direction there are two suspected connections of particular note. First, whether or not the fundamental group of a space is isomorphic to a group of order-preserving deformations of the real line is suspected to be connected with foliations (that is, how to "tightly pack" a space with lower--dimensional spaces). This connection is already understood in a select few cases. It is also conjecturally connected to Heegaard--Floer homology, a powerful new homology theory that has been used over the past decade to resolve many long-standing open questions in the study of 3-dimensional spaces. Together, this package of suspected connections has become known as "the L-space conjecture". One of the primary short-term goals of this program is the development and application of algebraic tools to tackle certain cases of the L-space conjecture. Advances in this direction contribute directly to the ongoing effort in the low-dimensional topology community to relate modern tools and techniques (such as Heegaard-Floer homology) to classical invariants and topological constructions from algebraic topology.

群体是从对称和僵化动作的研究开发的数学对象，尽管在现代研究中，在许多地方，小组作为一种编码某些数据的复杂方式出现。因此，挑战之一是学习如何从对其代数特性的研究中提取组编码的数据。 应对这一挑战的一种方法是表明您的兴趣群是（与）另一组的子组同构（相同），该小组是更好地理解的。我在研究中采用了这种方法，其中一个共同的主题是试图实现一个给定的群体作为低维空间（即低维物体的变形集合）（例如真实线或圆圈）的一组同构同态的子组。更一般而言，有时将小组作为订单保留功能集合从任意有序设置的集合中很有用。然后，该程序的目的是研究一个不同的方式，即可以将组作为真实线，圆或有序集的变形子组实现，并从这些结构中提取有关给定组的代数信息。当一个人通过称为“基本组”的结构研究群体的研究中，这是一个尤为重要的情况。在这种情况下，目标是从有关基础空间拓扑的基本小组信息中提取的目标；在这个方向上，有两个可疑的特定连接。首先，是否怀疑认为，空间的基本组是否与一组订单保留的变形构成同构，以免与叶子相连（也就是说，如何“紧紧打包”一个具有较低维空间的空间）。在几种情况下，已经了解了此连接。它也与Heegaard（Floer同源性）相关，这是一种强大的新同源性理论，在过去十年中已被用于解决三维空间研究中的许多长期开放问题。总之，这种可疑连接的包装已被称为“ L空间猜想”。该计划的主要短期目标之一是开发和应用代数工具来解决L空间猜想的某些情况。在这个方向上的进步直接有助于低维拓扑界的持续努力，以将现代工具和技术（例如Heegaard-loer同源性）与代数拓扑结构的古典不变性和拓扑结构联系起来。