Mapping Class Groups and Transformation Groups

映射类组和转换组

• 批准号: 2203178
• 负责人: Lei Chen
• 金额: $ 13.28万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203178&HistoricalAwards=false
• 关键词: Mapping; Class; Groups; Transformation

This project is a multi-year research plan to develop new tools for the study of the structure of transformation groups and mapping class groups. The transformation groups such as diffeomorphism groups and homeomorphism groups of a manifold describe all symmetries of the manifold. A manifold is a space that is locally a Euclidean space. The surface of the earth, the surface of a donut and our three or four dimensional universe are all examples of manifolds. The mapping class group is a reduced form of the homeomorphism group of a manifold, in which two transformations are identified if they can be connected by a path of transformations. These objects have connections with many areas of mathematics, including geometric topology, geometric group theory, dynamics, number theory, quantum field theory, representation theory, and algebraic geometry. This project studies a number of questions about the structure of symmetries of manifolds. Broader impacts of these efforts include reading groups for graduate students. Previously, in a joint work with Kathryn Mann, the principal investigator has proved Ghys' dimension conjecture that given two manifolds M and N, if the homeomorphism group of M has a nontrivial homomorphism to the homeomorphism group of N, then the dimension of M must be less than or equal to the dimension of N. To prove this, an "Orbit Classification Theorem" is developed, which says that every orbit of such a homomorphism is homeomorphic to a cover of some configuration space of M. This finding has a potential to give more results about classifying homomorphisms between transformation groups. Since we have figured out all orbits, now the challenge is to understand how to glue those orbits inside the manifold N. Another line of work studies the projection map from the transformation group to the mapping class group. In particular, we ask which subgroups of the mapping class group have sections under the projection, where the existence of sections implies that the surface bundle that the subgroup determines will admit a flat structure. In previous work on this subgroup question the principal investigator has proved several results using techniques of "hidden torsion" and rotation number. More tools will be developed specifically to attack this question. Another direction to explore concerns diffeomorphism groups of high dimensional manifolds, specifically, whether there is an example of an infinite torsion group that acts faithfully on some manifold.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目是一项为期多年的研究计划，旨在开发新工具，用于研究转型组和映射课程组的结构。诸如多态性群体和同构群等转化组描述了流形的所有对称性。歧管是局部欧几里得空间的空间。地球表面，甜甜圈的表面和我们的三个或四个维度宇宙都是歧管的例子。映射类组是歧管同态组的降低形式，在该组中，如果可以通过转换路径连接两个转换，则可以确定两个转换。 这些对象与数学的许多领域都有联系，包括几何拓扑，几何群体理论，动力学，数理论，量子场理论，表示理论和代数几何形状。该项目研究了许多有关歧管对称性结构的问题。 这些努力的更广泛的影响包括为研究生阅读小组。以前，在与凯瑟琳·曼恩（Kathryn Mann）的共同合作中，主要研究者证明了ghys的尺寸构想，如果m和n同构的同构群具有非平凡的同构同性形态，则对n的同型同构群体的同态构态，那么m的维度必须小于或等于n。同构对M的某些配置空间的封面是同构的，这一发现有可能在转换组之间对同态分类产生更多结果。由于我们已经弄清楚了所有轨道，因此现在的挑战是了解如何将这些轨道粘合在歧管N中。另一个工作线研究了从转换组到映射类组的投影图。特别是，我们询问映射类组的哪些亚组在投影下有部分，其中部分的存在意味着子组确定的表面束将允许平坦的结构。在此亚组问题的先前工作中，主要研究者使用“隐藏扭转”和旋转编号证明了几个结果。将专门开发更多工具来攻击这个问题。探索问题的另一个方向是，高维流形的差异性群体，具体是，是否有一个无限扭转组的例子，该组是否忠实于某种歧管，这反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。