Classifying 4-manifolds

4-流形分类

• 批准号: EP/T028335/1
• 负责人: Mark Powell
• 金额: $ 45.88万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT028335%2F1
• 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.

流形是一个局部欧几里得的拓扑空间，即在每个小邻域内看起来像欧几里德空间R^n，对于某个n，数n是流形的维度。拓扑学中最基本的问题之一是对流形进行分类。为了使这个问题更容易处理，我们通常局限于紧凑的、连通的流形；粗略地说，那些流形的大小是有界的，并且每两个点之间都有一条路径。每一个紧凑的、连通的一维流形都等价于或同胚于一个圆。曲面，或二维流形，在19世纪被分类。我们有可定向曲面和不可定向曲面，其中可定向曲面具有一些非负数量的孔，通过添加句柄从球面获得，不可定向曲面通过向球面添加Möbius带来获得。值得注意的是，在过去的50年里，人们对3维流形的理解相当好，由于瑟斯顿、佩雷尔曼和Agol取得了重要的突破。另一方面，斯梅尔的h-余边定理，Kervaire-Milnor的奇异球面，以及Browder-Novikov-Sullivan-Wall的外科手术程序，都导致了对至少5维流形的同样深刻的理解，尽管仅限于特殊类型的流形。这项工作帮助斯梅尔、米尔纳、诺维科夫、沙利文和瑟斯顿赢得了菲尔兹奖牌。在高维和低维流形拓扑的交汇处，4维流形占据了一个奇怪的中间地带。来自高维和低维流形的许多技术都部分地扩展到了四维，但到目前为止还没有确定的结果。因此，杰出的谜团比比皆是。例如，光滑的Poincaré猜想，每一个同伦的4-球面都与4-球面同构，舍恩费尔斯问题，每一个3-球面在4-球面上的光滑嵌入都与标准的赤道嵌入同构。另一方面，从低维几何方法，如纽结理论，高维外科理论，群论和数学物理，以及专门针对4维的技巧，有丰富的方法来研究4-流形。特别是Freedman和Donaldson的Fields勋章工作，打开了4-流形的世界。该项目旨在通过用代数不变量对4维流形进行分类来提高我们对4维的理解。在给定两个4-流形的情况下，我们寻找可计算的不变量来决定两个4-流形是否相同，类似于二维曲面上的空洞数目。我已经在这个方向上确定了一些悬而未决的问题，我相信，以我的专业知识，这些问题是可以处理的。特别是，某些具有所谓循环基本群的4-流形还没有被很好地理解，但如果有足够的工作，这应该是可能的。四维流形有两种截然不同的风格：光滑和拓扑。粗略地说，光滑流形允许使用可微函数来描述，而拓扑流形可以稍微狂野一些。该项目的重点是拓扑流形。通常可以得到关于拓扑4-流形的完整结果，因为它们可以表现出与代数更精确的对应，而没有类似的全局程序来理解它们的光滑近亲。

项目成果

SIMPLY CONNECTED MANIFOLDS WITH LARGE HOMOTOPY STABLE CLASSES

• DOI: 10.1017/s1446788722000167
• 发表时间: 2021-09
• 期刊: Journal of the Australian Mathematical Society
• 影响因子: 0.7
• 作者: Anthony Conway;D. Crowley;Mark Powell;Joerg Sixt

Stably diffeomorphic manifolds and modified surgery obstructions

稳定微分同胚流形和改进的手术障碍

• DOI: 10.48550/arxiv.2109.05632
• 发表时间: 2021
• 作者: Conway A

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

$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

$4$-具有边界和基本群 $mathbb{Z}$ 的流形

• DOI: 10.48550/arxiv.2205.12774
• 发表时间: 2022
• 作者: Conway A

Characterisation of homotopy ribbon discs

同伦带盘的表征

• DOI: 10.1016/j.aim.2021.107960
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Conway A