Topology of Four-Manifolds

摘要本课题研究四维流形的拓扑结构。最突出的问题是四维s-协定定理的良好基本群的表征，四维（同调）手术理论的公式，以及拓扑结谐和群的理解。本文将现有的幂零/可解群论和4流形中2球和惠特尼塔的高阶交数的方法与l2同调和冯·诺伊曼代数的方法相结合。近几十年来最令人兴奋的数学发现之一是欧几里得空间的表现非常不同，这取决于维度是否等于4（这是我们生活的维度，允许时间作为第四维）。更准确地说，除了n=4之外，有一种独特的方法来做核n空间的微积分，因为n=4有无数这样的理论。这些是欧几里得4空间上的“奇异”结构，由唐纳森、弗里德曼等人在1983年左右发现。这个项目在更复杂的四维流形上处理类似的问题。例如，它们可能有非平凡的基本群，所以它们可能比欧几里得四维空间更类似于我们的宇宙。在这种情况下，基本问题仍然是开放的，似乎与冯诺伊曼在量子力学工作中引入的“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.