Topology of Four-Manifolds

建议：DMS-0072775PI：Peter Teichner摘要本项目研究4维流形的拓扑。最突出的问题是四维S-余边定理好的基本群的刻画，四维(同调)外科理论的建立，以及对拓扑纽结协调群的理解。4-流形中2-球和惠特尼塔的幂零/可解群理论和高阶交数的现有技术将与L2-同调和von Neumann代数的方法相结合。过去几十年来最令人兴奋的数学发现之一是，欧几里德空间的行为非常不同，取决于维度等于还是不等于4(这是我们生活的维度，允许时间作为第四维)。更准确地说，在欧几里得n-空间中有一种独特的方法来计算微积分，除了n=4，在那里有无数这样的理论。这些是由Donaldson、Freedman等人在1983年左右发现的欧几里得4维空间上的“奇异”结构。这个项目处理更复杂的4维流形上的类似问题。例如，它们可能有非平凡的基本群，所以它们可能比欧几里德4维空间更接近我们的宇宙。在这种情况下，基本问题仍然是悬而未决的，似乎与冯·诺伊曼在他的量子力学著作中引入的“连续维度”有很深的关系。

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.