Non-Commutative Algebraic Phenomena in the Topology of Three- and Four-dimensional Spaces

三维和四维空间拓扑中的非交换代数现象

• 批准号: 0104275
• 负责人: Tim Cochran
• 金额: $ 24.68万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104275&HistoricalAwards=false
• 关键词: Non; Commutative; Algebraic; Phenomena; Topology

AbstractAward: DMS-0104275Principal Investigator: Tim CochranThis project develops a new area of noncommutative algebraictopology and its applications to low-dimensional topology. Thesuccess of algebraic topology in knot theory, for example, has,until recently centered around abelian invariants, that is tosay, invariants associated to the universal abelian coveringspace of the knot or link exterior. These invariants are theAlexander module, which is the first homology of this cover as amodule over a commutative Laurent polynomial ring, and theBlanchfield pairing. These determine the S-equivalence class ofthe knot as well as various other invariants. From theperspective of the knot group G, the Alexander module is simplyG'/G". Hence any behavior associated to G" will be invisible tothese abelian invariants. We remedy this deficiency by studyingthe quotients of successive terms of the higher derived series,or, put another way, study modules associated to more generalsolvable covering spaces. These are modules over noncommutativerings and thus are difficult to work with. We use techniques fromnoncommutative algebra and C* algebras to define invariants. Wefind , for each integer n, an entire theory which parallels theAlexander module and Blanchfield form and signatures. There areapplications to estimating genus, detecting fibered knots and3-manifolds, new invariants of concordance and representations ofmapping class groups.The advent of quantum mechanics led scientists to manyparadoxical, but now accepted, facts about our universe. Inparticular, there came the realization that "commutativemathematics" was inadequate to describe our physicalworld. Recall that 2 times 3 equals 3 times 2 is the commutativelaw of multiplication of numbers. Quantum mechanics showed thatphysical quantities are not mere numbers but more like arrays ormatrices of numbers. Since multiplication of matrices is notcommutative, this explains and models noncommutative phenomena atthe most fundamental levels of the physical world. This projectstudies the shape of 3 and 4-dimensional spaces by using newnoncommutative mathematics arising from algebra.

摘要奖项：DMS-0104275 首席研究员：Tim Cochran 该项目开发了非交换代数拓扑学的新领域及其在低维拓扑中的应用。例如，直到最近，结理论中代数拓扑的成功一直集中在阿贝尔不变量上，也就是说，与结或链接外部的通用阿贝尔覆盖空间相关的不变量。这些不变量是亚历山大模（这是此封面作为交换洛朗多项式环上的模的第一个同源性）和布兰奇菲尔德配对。这些决定了结的 S 等价类以及各种其他不变量。从结群 G 的角度来看，亚历山大模就是 G'/G"。因此，任何与 G" 相关的行为对于这些阿贝尔不变量来说都是不可见的。我们通过研究更高阶派生级数的连续项的商来弥补这一缺陷，或者换句话说，研究与更一般可解的覆盖空间相关的模块。这些是非交换环上的模块，因此很难使用。我们使用非交换代数和 C* 代数的技术来定义不变量。我们发现，对于每个整数n，都有一个与亚历山大模块和布兰奇菲尔德形式和签名相似的完整理论。这些应用可用于估计属、检测纤维结和三流形、新的一致性不变量以及映射类群的表示。量子力学的出现使科学家们发现了许多关于我们的宇宙的自相矛盾但现在已被接受的事实。特别是，人们认识到“交换数学”不足以描述我们的物理世界。回想一下，2 乘以 3 等于 3 乘以 2 是数字乘法的交换律。量子力学表明，物理量不仅仅是数字，而更像是数字的数组或矩阵。由于矩阵乘法不满足交换律，因此这可以在物理世界的最基本层面上解释和模拟非交换现象。该项目通过使用代数中产生的新的非交换数学来研究 3 维和 4 维空间的形状。