Symmetries of 4-manifolds

4-流形的对称性

• 批准号: EP/V04821X/2
• 负责人: Mark Powell
• 金额: $ 14.58万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV04821X%2F2
• 关键词: Symmetries; manifolds

A manifold is a topological space that is locally euclidean, that is in every small neighbourhood looks like euclidean spaceR^n, for some n. The number n is the dimension of the manifold. One of the most fundamental questions in topology is toclassify manifolds. In order to make the question more manageable, we often restrict to compact, connected manifolds;those that roughly speaking are of bounded size, and every two points has a path between them. Every compact,connected 1-dimensional manifold is equivalent, or homeomorphic, to a circle. Surfaces, or 2-dimensional manifolds, wereclassified in the 19th century. We have the orientable surfaces with some nonnegative number of holes, obtained from thesphere by adding handles, and nonorientable surfaces obtained by adding Möbius bands to the sphere instead.Remarkably, manifolds of dimension 3 have been understood rather well in the last 50 years, with important breakthroughsdue to Thurston, Perelman and Agol. On the other hand the h-cobordism theorem of Smale, exotic spheres of Kervaire-Milnor, and the surgery programme of Browder-Novikov-Sullivan-Wall, led to a likewise deep understanding of manifolds ofdimension at least 5. This work helped Smale, Milnor, Novikov, Sullivan, and Thurston win Fields medals.Manifolds of dimension 4 occupy a curious middle ground, at the confluence of high and low dimensional manifoldtopology. Many techniques from both high and low dimensional manifolds partially extend to dimension four, but thus farnever conclusively.As a result, outstanding mysteries abound. For example, the smooth Poincaré conjecture that every homotopy 4-sphere isdiffeomorphic to the 4-sphere, the Schoenflies problem that every smooth embedding of the 3-sphere in the 4-sphere isisotopic to the standard equatorial embedding remain open.On the other hand there are a wealth of techniques for studying 4-manifolds, coming from low dimensional geometric methods such as knot theory, high dimensional surgery theory, group theory and mathematical physics, as well as techniques special to dimension 4. In particular the Fields medal work of Freedman and Donaldson opened up the world of 4-manifolds.The aim of this project is to understand symmetries of 4-manifolds: a symmetry of a manifold is a self-map that preserves the structure. These are called homeomorphisms, or in the case of smooth manifolds, they are called diffeomorphisms. To avoid repeating myself, let me just discuss homeomorphisms from now on; everything I say has an analogue for diffeomorphims. The set of homeomorphisms from a manifold to itself form a group, and they also form a topological space in a natural way. This means that one can study the set of homeomorphisms from the point of view of group theory and of algebraic topology. The most basic question is to determine when two homeomorphisms are isotopic, meaning that one map can be continuously deformed until it agrees with the other map. The isotopy classes of homeomorphisms of a manifold also form a group, called the mapping class group of the manifold. Studying these groups for surfaces is both an old, beautiful topic, and the subject of significant current research. It is an exciting new area to investigate the analogous question for 4-dimensional manifolds. The principal goal of this project is to develop new machinery and techniques with which to do so, and to make new computations of 4-dimensional mapping class groups.

歧管是一个局部欧几里得的拓扑空间，在每个小社区中看起来像欧几里得的间隔^n，对于某些n。数字n是歧管的维度。拓扑中最根本的问题之一是分类流形。为了使问题更易于管理，我们经常限于紧凑的，连接的歧管；那些大致说明的大小是有限的，每两个点之间都有一个路径。每个紧凑的，连接的一维歧管都是等效的或同构的，都与一个圆相等。表面或二维流形，在19世纪被归类。我们具有可定向的表面，并具有一些非负孔的孔，通过添加手柄从TheSphere获得，而通过将Möbius乐队添加到球体中获得的不可方向的表面，而不是在过去的50年中，在过去的50年中相当好，并以重要的突破性的突破性地了解了TheSthofe Spherthroughsdue，perelman and perelman and perelman and perelman and agol man and perelman and perelman and perelman and aglman and perel。另一方面，Smale的H-Cobordism定理，Kervaire-Milnor的异国情调领域以及Browder-Novikov-Sullivan-Wall的手术计划，导致对维度的多种流形有深刻的了解。这项工作有助于这项工作。汇合处的许多高度和低维歧管的技术部分扩展到了第四尺寸，但最终范围是最终的。结果，杰出的神秘之神很多。例如，平稳的庞加莱猜想是，每个同质镜4球质量异常对四个球员，schoenflies的问题，即在4个球体中的每一个平滑嵌入在4个球体中的平滑嵌入到标准等效嵌入中的既定等效性仍然是开放的。另一方面，有一个较低的技术，即从较低的理论中进行4-元素的范围，从而涉及4-元素的范围。维度手术理论，群体理论和数学物理学以及对维度4的技术。特别是Freedman和Donaldson的领域医学工作开放了4个manifold的世界。该项目的目的是理解4个manifolds的对称性：A-Manifolds的对称性：A歧管的对称性是一种自我映射，可以保留结构。这些被称为同构，或者在光滑的歧管的情况下，它们被称为差异性。为了避免重复自己，让我从现在开始讨论同构的同态；我所说的一切都具有差异性的类似物。同态从流形到本身的一组同态形成，它们也以自然的方式形成了拓扑空间。这意味着人们可以从群体的理论和代数拓扑的角度研究一组同构形态。最基本的问题是确定何时两个同态是同位素，这意味着可以连续变形一个地图，直到它与另一个地图一致为止。歧管同态同态的同位素类也形成了一个组，称为歧管的映射类组。研究这些群体的表面既是一个古老的，美丽的话题，也是当前重大研究的主题。这是一个令人兴奋的新领域，可以研究四维流形的类似问题。该项目的主要目标是开发使用的新机械和技术，并对4维映射课程组进行新的计算。

项目成果

Smoothing 3-manifolds in 5-manifolds

平滑 5 流形中的 3 流形

• DOI: 10.48550/arxiv.2309.15962
• 发表时间: 2023
• 作者: Daher M

Infinite homotopy stable class for 4-manifolds with boundary

带边界的 4 流形的无限同伦稳定类

• 发表时间: 2023
• 期刊: Pacific journal of mathematics
• 影响因子: 0.6
• 作者: Conway, Anthony;Crowley Diarmuid, Powell Mark.
• 通讯作者: Crowley Diarmuid, Powell Mark.