4-manifolds and controlled topology

AbstractAward：DMS-0103976首席研究员：Frank Quinquiry项目涉及两个不同的领域：4维流形和高维受控拓扑。 最近的成功与拓扑4-流形建议的时间是成熟的，试图制定手术障碍，为任意的基本群体。对四维空间中物体结构的进一步理解和拓扑场论的新构造为光滑四维流形的组合定义不变量提供了希望。在控制区最近的工作与Ranicki提供了一个难以捉摸的稳定性定理控制手术障碍的证明。这可能会指出一个全面的描述群的方式，作为广义的同源性，类似于较早的描述结束和控制的h-cobor dismobilization。这样的描述将有应用范围从局部结构分层集锐化2-torsion结论在某些情况下的诺维科夫猜想。几何学和分析（连续的观点）揭示了四维空间的奇怪行为。用物理学中的数学术语来说，其他维度是“经典”的，而第四维似乎是“量子”的。拓扑学上，四维物体是用三维积木（一种离散的观点）来描述的，特别是三维空间中的结点和链接。一个目标是弥合这两种观点之间的差距，特别是从拓扑学的角度理解量子行为。在更高的维度中，拓扑学和定性拓扑学（例如外科手术理论）是离散的观点。二十年前，PI在一种情况下弥合了这与局部连续拓扑之间的差距，“伪合痕”。“这已经有了许多应用。更多的应用等待着这座桥梁的拓宽，以包括其他情况，例如代数K理论和外科手术，但这些方法是复杂的，技术性很强，迄今为止一直抵制扩展。该项目的另一个目标是贯彻这些概括。

AbstractAward: DMS-0103976Principal Investigator: Frank QuinnThis project involves two distinct areas: 4-dimensionalmanifolds, and high-dimensional controlled topology. Recentsuccesses with topological 4-manifolds suggest the time is ripeto try to formulate surgery obstructions for arbitraryfundamental groups. Improved understanding of handlebodystructures and a new construction of topological field theoriesin dimension 4 offer hopes for combinatorially-defined invariantsof smooth 4-manifolds. In the controlled area recent work withRanicki has provided a proof of an elusive stability theorem forcontrolled surgery obstructions. This may point the way to a fulldescription of the groups as generalized homology, analogous toearlier descriptions of the end and controlled h-cobordismobstructions. Such a description would have applications rangingfrom local structure in stratified sets to sharpening 2-torsionconclusions in some cases of the Novikov conjecture.This project explores the boundary between the discrete andcontinuous worlds. Geometry and analysis (continuous points ofview) have revealed strange behavior in dimension 4. In terms ofthe mathematics used in physics, other dimensions are "classical"while dimension 4 seems to be "quantum." Topologically4-dimensional objects are described in terms of 3-dimensionalbuilding blocks (a discrete point of view), particularly knotsand links in 3-space. One objective is to bridge the gap betweenthese two views, and in particular understand the quantumbehavior from the topological point of view. In higher dimensionsalgebraic and qualitative topology (e.g. surgery theory) arediscrete points of view. Twenty years ago the PI bridged the gapbetween this and local continuous topology in one case,"pseudoisotopy." This has had numerous applications. Many moreapplications await a widening of this bridge to include othercases, for instance algebraic K-theory and surgery, but themethods are complex and intensely technical and have so farresisted extension. The other objective of the project is tocarry through these generalizations.