Symmetries of 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流形的分类。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重新分类。我们有定向表面与一些非负数目的洞，从theschem获得通过添加处理，和nonorientable表面获得通过添加莫比乌斯带spheriainstead. Notably，流形的3维已被理解，而在过去的50年，与重要的突破，由于瑟斯顿，佩雷尔曼和Agol。另一方面的h-协边定理的Smale，异国情调领域的Kervaire，米尔诺，和手术方案的Browder，诺维科夫-Sullivan墙，导致了同样深刻的理解流形的尺寸至少5。这项工作帮助Smale，Milnor，Novikov，Sullivan和Thurston赢得了菲尔兹奖。四维流形占据了一个奇怪的中间地带，在高、低维流形拓扑的汇合处。许多从高、低维流形得到的技术都部分地延伸到了四维，但却永远无法得到结论性的结果，因此，仍然有许多未解之谜。例如，光滑Poincaré猜想（即所有同伦4-球面同构于4-球面），Schoenflies问题（即所有3-球面在4-球面中的光滑嵌入同构于标准赤道嵌入）仍然是一个未解决的问题。另一方面，研究4-流形的方法也非常丰富，这些方法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。特别是菲尔兹奖的工作弗里德曼和唐纳森开辟了世界的4流形。这个项目的目的是了解对称性的4流形：一个对称的流形是一个自我地图，保持结构。这些被称为同胚，或者在光滑流形的情况下，它们被称为非同胚。为了避免重复，从现在开始我只讨论同胚;我所说的每一个都有一个类似的同胚。从流形到其自身的同胚集合形成一个群，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的集合。最基本的问题是确定两个同胚何时是同位素的，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。流形的同胚的合痕类也形成一个群，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，也是当前重要研究的主题。研究四维流形的类似问题是一个令人兴奋的新领域。这个项目的主要目标是开发新的机器和技术，并进行新的计算4维映射类组。

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.