Symmetries of 4-manifolds

流形是一个局部欧几里得的拓扑空间，也就是说，对于某个n，在每个小邻域中看起来像欧几里德空间^n，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世纪被归类。我们有一些具有非负数量的孔的可定向曲面，通过添加句柄从球体中获得，以及通过将Möbius带添加到球体而获得的不可定向曲面。值得注意的是，在过去的50年里，人们对3维流形的理解已经相当好，由于瑟斯顿、佩雷尔曼和Agol的重要突破。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，导致了对至少5维流形的同样深刻的理解。这项工作帮助Smear，Milnor，Novikov，Sullivan和瑟斯顿获得了Fields奖章。4维的流形占据了一个奇怪的中间地带，在高维和低维流形拓扑的交汇处。来自高维和低维流形的许多技术都部分地扩展到了四维，但到目前为止都不是决定性的。因此，杰出的奥秘比比皆是。例如，光滑的Poincaré猜想，每个同伦的4-球面都微分同构于4-球面，每个3-球面在4-球面上的光滑嵌入到标准的赤道嵌入保持开放。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，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维流形的世界。这个项目的目的是了解4维流形的对称性：流形的对称是保持结构的自映射。它们被称为同胚，或者在光滑流形的情况下，它们被称为微分同胚。为了避免重复我自己，让我从现在开始只讨论同胚；我所说的每一件事都有一个关于微分同胚的类比。从流形到流形本身的一组同胚构成一个群，它们也以自然的方式形成一个拓扑空间。这意味着人们可以从群论和代数拓扑学的角度来研究同胚集。最基本的问题是确定两个同胚何时是同位的，这意味着一个映射可以连续变形，直到它与另一个映射一致。流形的同胚类也构成一个群，称为流形的映射类群。研究这些群体的表面既是一个古老而美丽的话题，也是当前重大研究的主题。研究四维流形的类似问题是一个令人振奋的新领域。这个项目的主要目标是开发新的机械和技术来做到这一点，并对4维映射类群进行新的计算。

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.