Mapping Class Groups and Transformation Groups

该项目是一个多年研究计划，旨在开发新的工具，用于研究转换组和映射类组的结构。流形上的变换群，如同胚群和自同构群，描述了流形的所有对称性。一个流形是一个局部为欧几里得空间的空间。地球的表面、甜甜圈的表面以及我们的三维或四维宇宙都是流形的例子。映射类群是流形的同胚群的一个约化形式，其中两个变换是可识别的，如果它们可以通过变换的路径连接。 这些对象与数学的许多领域有联系，包括几何拓扑学、几何群论、动力学、数论、量子场论、表示论和代数几何。本课题研究了流形对称性结构的若干问题。 这些努力的更广泛影响包括研究生阅读小组。在此之前，在与凯瑟琳·曼（Kathryn Mann）的合作中，主要研究者证明了盖斯的维数猜想，即给定两个流形M和N，如果M的同胚群与N的同胚群有非平凡同态，则M的维数必须小于或等于N的维数。为了证明这一点，提出了一个“轨道分类定理”，即这样一个同态的每个轨道都同胚于M的某个构形空间的一个覆盖。这一发现有可能给出更多关于变换群间同态分类的结果。既然我们已经计算出了所有的轨道，现在的挑战是理解如何将这些轨道粘在流形N内。 另一种工作是研究从转换组到映射类组的投影映射。特别是，我们问哪些子群的映射类组有部分下的投影，其中部分的存在意味着，表面丛的子群确定将承认一个平坦的结构。在以前的工作中，这个小组的问题，主要研究者已经证明了几个结果，使用技术的“隐藏扭转”和旋转数。将专门开发更多的工具来解决这个问题。另一个探索的方向是高维流形的同构群，具体来说，是否存在一个无限扭转群忠实地作用于某些流形的例子。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.