Mathematical Sciences: Topology of 4-Manifolds

9703996 Teichner这个项目处理与4维流形有关的拓扑问题。要回答的主要问题是，高维外科技术和S-余波德定理对哪些基本群起作用。现有的技术包括复杂的卡森句柄结构，可以肯定地回答一大类群体的问题。此外，还提出了一种寻找自由基本群的障碍的幂零群理论方法。这两种方法也将应用于4维空间中不相交的2-球面的链环同伦分类，以及3维空间中关于纽结和链环的调和类问题。过去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，取决于维度n等于还是不等于4(这是我们生活的维度，允许时间作为第四维度)。更确切地说，我们都学习了一个实变量的微积分，我们中的一些人继续学习了n个变量的微积分，即欧几里得n-空间上的光滑函数理论。但很少有人被告知，对于n=4，存在(数不清的！)许多这样的理论，而对于所有其他维度，有一个独特的，那就是我们被教授的那个。用数学术语来说，我指的是欧几里德4-空间中的“奇异”结构，由唐纳森、弗里德曼等人在1983年前后发现。这个项目处理更复杂的4维流形(从技术上讲，那些具有非平凡基本群的流形)上的类似问题。与欧几里得4-空间的一个具体区别是，在一般的4-流形中，并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. ***