Mathematical Sciences: Topology of 4-Manifolds

小行星9703996 这个项目处理与四维流形有关的拓扑问题。 需要回答的主要问题是手术和s配边定理的高维技术适用于哪些基本群体。 现有的技术包括复杂的卡森处理，回答了肯定的问题，为一大类群体的建设。 此外，还有一个建议的幂零群论方法定位障碍的自由基本群。 这两种方法也将适用于链接同伦分类不相交的2-领域在4-空间，以及有关问题的协调类的结和链接在3-空间。 在过去的十年中，最令人兴奋的数学发现之一是欧几里得n空间的行为非常不同，这取决于n维是否等于4（这是我们生活的维度，允许时间作为第四维）。 更准确地说，我们都学习过一个真实的变量的微积分，我们中的一些人继续学习n个变量的微积分，即，欧氏空间上的光滑函数理论。 但很少有人被告知，对于n=4，有（不可数！）许多这样的理论，而对于所有其他维度，有一个独特的理论，即我们被教导的理论。 用数学术语来说，我说的是欧几里得四维空间上的“奇异”结构，它是由唐纳森、弗里德曼和其他人在1983年左右发现的。 这个项目处理更复杂的四维流形（技术上，那些有非平凡的基本群）上的类似问题。 与欧几里得四维空间的一个具体区别是，在一般的四维流形中，不是每个向量场都对应于势能函数。 ***

9703996 Teichner This project deals with topological problems related to 4-dimensional manifolds. The main question to be answered is for which fundamental groups the higher dimensional techniques of surgery and s-cobordism theorems work. The existing techniques include sophisticated constructions of Casson-handles that answer the question in the affirmative for a large class of groups. In addition, there is a proposed nilpotent group theory method for locating obstructions for free fundamental groups. Both methods will also be applied to the link homotopy classification of disjoint 2-spheres in 4-space as well as to questions concerning concordance classes of knots and links in 3-space. One of the most exciting mathematical discoveries in the last decade is the fact that Euclidean n-space behaves very differently depending on whether the dimension n equals or doesn't equal 4 (which is the dimension we live in, allowing time as the fourth dimension). More precisely, we have all learned calculus in one real variable, and some of us went on to learn calculus in n variables, i.e., the theory of smooth functions on Euclidean n-space. But very few of us were told that for n=4 there are (uncountably!) many such theories, whereas for all other dimensions there is a unique one, namely the one we were taught. In mathematical terms, I am talking about the "exotic" structures on Euclidean 4-space, discovered by Donaldson, Freedman and others around 1983. This project deals with similar questions on more complicated 4-dimensional manifolds (technically, those which have nontrivial fundamental group). One concrete difference from Euclidean 4-space is that in a general 4-manifold, not every vector-field corresponds to a potential energy function. ***