Topology of Four-Manifolds

提案：DMS-0072775 PI：Peter Teichner摘要本项目研究四维流形的拓扑结构。最突出的问题是良好的基本群体的4维s-协边定理，制定了4维（同源）手术理论的特征，和理解的拓扑结和谐组。 将幂零/可解群理论和4-流形中2-球面和Whitney塔的高阶交数的现有方法与L2-同调和von Neumann代数的方法结合起来，在过去几十年中最令人兴奋的数学发现之一是欧几里得空间的行为非常不同，这取决于维数是否等于4（这是我们生活的维度，允许时间作为第四维）。更确切地说，在欧几里得n-空间中有一种独特的方法来做微积分，除了n=4，那里有无数这样的理论。这些是欧几里得四维空间上的“奇异”结构，由唐纳森、弗里德曼和其他人在1983年左右发现。这个项目处理更复杂的四维流形上的类似问题。例如，这些可能有非平凡的基本群，所以它们可能比欧氏四维空间更接近我们的宇宙。在这种情况下，基本问题仍然是开放的，似乎有一个深刻的关系，“连续维”介绍了冯诺依曼在他的工作量子力学。

Proposal: DMS-0072775PI: Peter TeichnerABSTRACTThis project studies the topology of 4-dimensional manifolds. Themost prominent problems are the characterization of goodfundamental groups for the 4-dimensional s-cobordism theorem, theformulation of a 4-dimensional (homology) surgery theory, and theunderstanding of the topological knot concordance group. Theexisting techniques from nilpotent/solvable group theory and higherorder intersection numbers of 2-spheres and Whitney towers in4-manifolds are to be combined with methods from L2-homology andvon Neumann algebras.One of the most exciting mathematical discoveries in the lastdecades is the fact that Euclidean space behaves very differentlydepending on whether the dimension equals or doesn't equal four(which is the dimension we live in, allowing time as the fourthdimension). More precisely, there is a unique way to do calculus inEuclidean n-space except for n=4 where there are uncountably many such theories. These are 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. For example, these mayhave nontrivial fundamental group, so they are possibly moresimilar to our universe than Euclidean 4-space. In this case thefundamental questions are still wide open and there seems to be adeep relation to the "continuous dimension" introduced by vonNeumann in his work on quantum mechanics.