Classifying 4-manifolds

流形是一个局部欧几里得的拓扑空间，也就是在每个小邻域中看起来像欧几里得空间R^n，对于某个n，数字n是流形的维数。拓扑学中最基本的问题之一是流形的分类。为了使这个问题更易于管理，我们经常限制使用紧致的、连通的流形；粗略地说，这些点的大小是有限的，每两个点之间都有一条路径。每一个紧的、连通的一维流形都等价于一个圆，或者说同胚的。曲面或二维流形在19世纪被分类。在球面上加入手柄得到具有非负孔数的可定向曲面，在球面上加入Möbius带得到不可定向曲面。值得注意的是，在过去的50年里，三维流形得到了很好的理解，Thurston、Perelman和Agol取得了重大突破。另一方面，kervair - milnor的小的、奇异球体的h协同定理，以及Browder-Novikov-Sullivan-Wall的手术程序，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、米尔诺、诺维科夫、沙利文和瑟斯顿赢得了菲尔兹奖。在高维流形和低维流形拓扑的交汇处，四维流形占据了一个奇特的中间地带。许多来自高维和低维流形的技术部分地扩展到四维，但到目前为止还没有定论。因此，悬而未决的谜团比比皆是。例如，光滑的poincar<s:1>猜想（每个同伦4球都与4球微分同构）、schoenfly问题（每个3球在4球中的光滑嵌入都与标准赤道嵌入同位素）仍然是开放的。另一方面，研究4-流形的技术非常丰富，包括低维几何方法，如结理论、高维外科理论、群论和数学物理，以及专门研究4维流形的技术。特别是弗里德曼和唐纳森的菲尔兹奖工作打开了4流形的世界。该项目旨在通过根据代数不变量对4-流形进行分类来提高我们对4维的理解。给定两个4-流形，我们寻求可计算的不变量，可以决定两个4-流形是否相同，类似于二维曲面上的孔数。在这个方向上，我已经确定了一些悬而未决的问题，我相信以我的专业知识，这些问题是可以解决的。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，但只要有足够的工作，这应该是可能的。四维流形有两种不同的风格：平滑的和拓扑的。粗略地说，光滑流形允许使用可微函数来描述，而拓扑流形则可以更狂野一些。该项目侧重于拓扑流形。通常可以得到拓扑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关系，而没有类似的全局程序来理解它们的光滑表兄弟。

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.