3-dimensional manifolds and related topic

3 维流形及相关主题

• 批准号: 0305846
• 负责人: Cameron Gordon
• 金额: $ 9万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305846&HistoricalAwards=false
• 关键词: dimensional; manifolds; related; topic

AbstractAward: DMS-0305846Principal Investigator: Cameron Gordon(1) The goal of the project is to investigate several problems inand around 3-dimensional topology. A major focus will be theCabling Conjecture, which asserts that Dehn surgery on ahyperbolic knot in the 3-sphere always yields a prime3-manifold. This is part of the program to completely describeall non-hyperbolic Dehn surgeries on hyperbolic knots. Recently,John Luecke and the principal investigator showed that thehyperbolic knots with non-integral toroidal Dehn surgeries areprecisely those described by Eudave-Munoz, and some of thetechniques developed there may be applicable to the CablingConjecture. Another part of the general program that will beaddressed is the conjecture that any Seifert fiber space surgeryon a hyperbolic knot must be integral. The project will alsoconsider various geometric group theoretic topics that arerelated to the theory of 3-manifolds, such as surface subgroupsof Coxeter and Artin groups, and Gromov's question as to whetheror not a 1-ended word hyperbolic group always has a surfacesubgroup. Another question concerning surfaces and 3-manifoldsthat will be investigated is the Simple Loop Conjecture, whichasserts that if a map from a closed orientable surface to aclosed orientable 3-manifold is not injective on fundamentalgroup, then there is an embedded essential loop in the surfacewhose image is null-homotopic in the 3-manifold.(2) The project is part of the general goal to understand thestructure of 3-dimensional manifolds. These are objects that arelocally like ordinary 3-dimensional space, but whose globalstructure may be quite complicated. Since we live in a3-manifold, one might say that 3-dimensional topology aims todescribe what the mathematical possibilities are for our spatialuniverse. One important aspect of 3-dimensional topology is thetheory of knots - a knot being a closed loop embedded somehow inspace. A wide variety of mathematical methods can be applied tothe study of knots, leading to new information about3-manifolds. Recently, deep connections between 3-dimensionaltopology and quantum physics were discovered in this way. Knottheory is related to the general theory of 3-manifolds through aconstruction known as Dehn surgery, in which a solid tube aroundthe knot is removed and sewn back in differently. We note that atheorem about Dehn surgery, the Cyclic Surgery Theorem, has beenused to determine the topological nature of the action of certainenzymes on strands of DNA. Many aspects of Dehn surgery are nowquite well understood. One of the main remaining questions, whichis a major focus of the project, is to show that (except in anobvious degenerate situation) the 3-manifold resulting from aDehn surgery on a knot never decomposes as a "sum" of two simplermanifolds. Another important tool in 3-dimensional topology isthe study of (2-dimensional) surfaces in 3-manifolds, and theproject will also address various questions in thisarea. Finally, the project will enable the principal investigatorto continue his involvement in the education and training ofgraduate students in topology at the University of Texas atAustin.

AbstractAward：DMS-0305846首席研究员：卡梅隆戈登（1）该项目的目标是研究三维拓扑中和周围的几个问题。一个主要的焦点将是电缆猜想，它声称，德恩手术的双曲结在3球总是产生一个素数3流形。这是程序的一部分，以完全描述所有非双曲Dehn手术的双曲结。 最近，John Luecke和主要研究者表明，具有非积分环形Dehn手术的双曲结正是Eudave-Munoz所描述的，并且那里开发的一些技术可能适用于电缆猜想。另一部分的一般程序，将被解决的是猜想，任何塞弗特纤维空间surgeron双曲结必须是积分。该项目还将考虑各种几何群论的主题，涉及到理论的3-流形，如表面subgroupsof Coxeter和Artin集团，和格罗莫夫的问题，是否或不1结束字双曲群总是有一个surfacesubgroup。 另一个与曲面和3-流形有关的问题是简单环猜想，它认为如果从一个闭可定向曲面到闭可定向3-流形的映射在基本群上不是内射的，则在该曲面中存在一个嵌入的本质环，该本质环的像在3-流形上是零同伦的。(2)该项目是理解三维流形结构的总体目标的一部分。这些物体在局部上就像普通的三维空间，但其整体结构可能相当复杂。 由于我们生活在三维流形中，有人可能会说三维拓扑旨在描述我们的空间宇宙的数学可能性。三维拓扑学的一个重要方面是纽结理论--纽结是以某种方式嵌入空间的闭合环。各种各样的数学方法可以应用到纽结的研究中，导致了关于三维流形的新信息。最近，三维拓扑学和量子物理学之间的深层联系就是以这种方式被发现的。纽结理论与三维流形的一般理论是通过被称为Dehn手术的构造联系在一起的，在Dehn手术中，一个围绕纽结的实心管被移除并以不同的方式缝回。我们注意到一个关于Dehn手术的定理，循环手术定理，已经被用来确定某些酶对DNA链作用的拓扑性质。 Dehn手术的许多方面现在已经很好地理解了。剩下的主要问题之一，这是该项目的一个主要焦点，是要表明（除了在一个明显的退化情况下）的3-流形所产生的aDehn手术结从来没有分解为两个simplermanifold的“总和”。三维拓扑学的另一个重要工具是三维流形中的（二维）曲面的研究，该项目也将解决这一领域的各种问题。最后，该项目将使首席执行官继续参与教育和培训研究生的拓扑在得克萨斯大学奥斯汀分校。