Classifying 4-manifolds

流形是一个局部欧氏的拓扑空间，也就是说，对于某个n，在每个小邻域中看起来都像欧氏空间R^n。数n是流形的维数。拓扑学中最基本的问题之一是对流形进行分类。为了使问题更易于处理，我们经常限制到紧凑的，连通的流形;粗略地说，那些有界的大小，每两个点之间有一条路径。每一个紧致的连通的一维流形都等价于或同胚于一个圆。曲面或二维流形在世纪被分类。我们有定向表面与一些非负数目的洞，从球获得的增加处理，和nonorientable表面获得的增加莫比乌斯带sphere instead. Notably，流形的3维已经了解，而在过去的50年里，重要的突破，由于瑟斯顿，佩雷尔曼和阿戈尔。另一方面的h-cobordism定理的Smale，异国情调领域的Kervaire，米尔诺，和手术方案的Browder，诺维科夫沙利文墙，导致了同样深刻的理解流形的维度至少5，虽然仅限于特殊类别的流形。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菲尔兹奖。四维流形占据了一个奇怪的中间地带，在高、低维流形拓扑的交汇处。许多从高、低维流形得到的技术都部分地延伸到了四维，但迄今为止还没有定论，因此，仍然存在许多未解之谜。例如，光滑Poincaré猜想（即所有同伦4-球面都是4-球面的同胚），Schoenflies问题（即所有3-球面在4-球面中的光滑嵌入都是标准赤道嵌入的同位素）仍然是一个悬而未决的问题。另一方面，研究4-流形的方法也是一个丰富的技术，这些技术来自于低维几何方法，如纽结理论，高维外科手术理论，群论和数学物理，以及专门的技术，以4维。特别是菲尔兹奖工作的弗里德曼和唐纳森开辟了世界4流形。该项目旨在通过对四维流形的代数不变量进行分类来提高我们对四维的理解。给定两个4-流形，我们寻找可以决定两个4-流形是否相同的可计算不变量，类似于二维曲面中的孔数。我已经确定了这方面的一些悬而未决的问题，我认为以我的专业知识，这些问题是可以解决的。特别是某些四维流形与所谓的循环基本群没有很好地理解，但有足够的工作，这应该是可能的。四维流形有两种不同的味道：光滑和拓扑。粗略地说，光滑流形可以用可微函数来描述，而拓扑流形可以稍微更怀尔德一些。该项目的重点是拓扑流形。拓扑4-流形的完整结果通常可以得到，因为它们可以表现出与代数更精确的对应，而没有类似的整体方案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.