Symmetries of 4-manifolds

流形是一个局部欧几里得的拓扑空间，它在每个小邻域中看起来都像欧几里得空间，对于某个n。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世纪被重新分类。在球面上通过添加手柄得到具有非负孔数的可定向曲面，在球面上添加Möbius带得到不可定向曲面。值得注意的是，在过去的50年里，由于Thurston、Perelman和Agol的重要突破，三维流形已经被很好地理解了。另一方面，kervire - milnor的小的、奇异的球体的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。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，即每一个同伦的4球与4球是微分同构的；schoenfly问题，即每一个3球的光滑嵌入在4球与标准赤道嵌入的同位素中保持开。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。这个项目的目的是理解4流形的对称性：流形的对称性是一个保持结构的自映射。这些被称为同胚，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，让我从现在开始讨论同胚；我说的每件事都和微分纯有相似之处。流形到自身的同胚集合构成一个群，它们也自然地构成一个拓扑空间。这意味着可以从群论和代数拓扑的角度研究同胚集。最基本的问题是确定什么时候两个同胚是同位素的，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。流形的同胚的同位素类也形成一个群，称为流形的映射类群。研究这些表面群既是一个古老而美丽的话题，也是当前重要的研究课题。研究四维流形的类似问题是一个令人兴奋的新领域。该项目的主要目标是开发新的机器和技术来实现这一目标，并对四维映射类组进行新的计算。

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.