Classifying 4-manifolds

4-流形分类

• 批准号: EP/T028335/2
• 负责人: Mark Powell
• 金额: $ 32.28万
• 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: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT028335%2F2
• 关键词: Classifying; manifolds

A manifold is a topological space that is locally euclidean, that is in every small neighbourhood looks like euclidean space R^n, for some n. The number n is the dimension of the manifold. One of the most fundamental questions in topology is to classify 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, were classified in the 19th century. We have the orientable surfaces with some nonnegative number of holes, obtained from the sphere 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 breakthroughs due 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 of dimension at least 5, albeit restricted to special classes of manifolds. 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 manifold topology. Many techniques from both high and low dimensional manifolds partially extend to dimension four, but thus far never conclusively.As a result, outstanding mysteries abound. For example, the smooth Poincaré conjecture that every homotopy 4-sphere is diffeomorphic to the 4-sphere, the Schoenflies problem that every smooth embedding of the 3-sphere in the 4-sphere is isotopic 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 project aims to improve our understanding of 4-dimensions by classifying 4-manifolds in terms of algebraic invariants. Given two 4-manifolds, we seek computable invariants that can decide whether two 4-manifolds are the same, analogous to the number of holes in a surface in dimension two. I have identified a number of open questions in this direction that I believe are tractable given my expertise. In particular certain 4-manifolds with so-called cyclic fundamental groups are not well understood, but with sufficient work this ought to be possible.Four dimensional manifolds come in two distinct flavours: smooth and topological. Roughly speaking, smooth manifolds admit a description using differentiable functions, whereas topological manifolds can be somewhat wilder. The project focusses on topological manifolds. Often complete results on topological 4-manifolds can be obtained, since they can exhibit a more precise correspondence with algebra, whereas there is no analogous global programme for understanding their smooth cousins.

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空间r^n，对于某些n。数字n是歧管的维度。拓扑结构中最基本的问题之一是对流形进行分类。为了使问题更易于管理，我们经常限于紧凑，连接的流形；那些大致说的是有限的大小，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。表面或二维流形被分类为19世纪。我们具有可定向的表面，并具有一些非负孔的孔，通过添加手柄从球体中获得，而通过将Möbius带添加到球体中获得的不可方向的表面，而不是瑟尔斯顿（Perelman）和Agolman和agolman和agolman和agolman和agolman构成的重要突破。另一方面，Smale的H-Cobordism定理，Kervaire-Milnor的外来领域以及Browder-Novikov-Sullivan-Wall的手术计划，至少使对尺寸的歧管的深刻理解至少5，尽管仅限于歧管的特殊类别。这项工作帮助Smale，Milnor，Novikov，Sullivan和Thurston Win Fields奖牌。维度4的歧管占据了一个好奇的中间立场，在高和低维歧管拓扑结构的汇合处。来自高维歧管的许多技术都部分扩展到了第四维度，但到目前为止，从未有结论。作为结果，杰出的神秘事件比比皆是。例如，平滑的庞加莱猜想是，每个同型4赛都与4赛季都有不同的形态，即四个球体中的每个平滑嵌入在4-Sphere中的每个平滑嵌入都是同位素对标准的应有嵌入同位素的同位素保持开放的同位素。维度手术理论，群体理论和数学物理学以及特殊的技术4。尤其是Freedman和Donaldson的领域医疗工作开放了4个Manifolds的世界。该项目旨在通过根据代数不变式对4个manifolds进行分类，以提高我们对4维的理解。给定两个4个manifolds，我们寻求可计算的不变式，可以决定两个4个manifolds是否相同，类似于二维中表面中的孔数。我已经在这个方向上确定了许多公开问题，鉴于我的专业知识，我认为这是可以处理的。尤其是某些具有所谓循环基本组的4个manifolds尚不清楚，但是在足够的工作中，这种巨大的歧管有两种不同的风味：光滑和拓扑。粗略地说，光滑的歧管承认使用可区分功能的描述，而拓扑歧管可能会有些荒谬。通常可以获得拓扑4 manifolds的完整结果，因为它们可以与代数表现出更精确的对应关系，而没有类似的全球计划来理解其光滑的表亲。

项目成果

Four-manifolds up to connected sum with complex projective planes

四流形直至复射影平面的连通和

• DOI: 10.1353/ajm.2022.0001
• 发表时间: 2022
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Kasprowski D

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

Counterexamples in 4-manifold topology

• DOI: 10.4171/emss/56
• 发表时间: 2022-03
• 期刊: EMS Surveys in Mathematical Sciences
• 影响因子: 2.3
• 作者: Daniel Kasprowski;Mark Powell;Arunima Ray

Embedded surfaces with infinite cyclic knot group

具有无限循环结组的嵌入表面

• DOI: 10.2140/gt.2023.27.739
• 发表时间: 2023
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Conway A

Embedding surfaces in 4-manifolds

将表面嵌入 4 流形

• DOI: 10.48550/arxiv.2201.03961
• 发表时间: 2022
• 作者: Kasprowski D