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

摘要奖：DMS-0104275主要研究人员：蒂姆·科克伦这个项目开拓了非对易代数拓扑及其在低维拓扑中的应用的一个新领域。例如，直到最近，纽结理论中的代数拓扑学的成功都是围绕着阿贝尔不变量，也就是说，与纽结或链外的泛阿贝尔覆盖空间相关的不变量。这些不变量是Alexander模和Blanchfield对，Alexander模是这个覆盖作为交换Laurent多项式环上的模的第一个同调。这些决定了纽结的S等价类以及其他各种不变量。从纽结群G的角度来看，Alexander模就是G‘/G“，因此与G”有关的任何行为对这些阿贝尔不变量都是不可见的。我们通过研究高阶导数的连续项的商来弥补这一不足，或者，换句话说，研究与更一般的可解覆盖空间相关的模。这些都是非对易运算上的模块，因此很难处理。我们使用非交换代数和C*代数中的技巧来定义不变量。对于每个整数n，我们发现了一个与Alexander模和Blanchfield形式和签名相对应的完整理论。量子力学在估计亏格、检测纤维结和3-流形、新的协调不变量和映射类群的表示方面都有应用。量子力学的出现导致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经被接受的事实。特别是，人们意识到“可交换的数学”不足以描述我们的物理世界。回想一下，2乘3等于3乘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，更像是数字的阵列或矩阵。由于矩阵的乘法是不可交换的，这解释并模拟了物理世界最基本的层面上的非交换现象。这个项目研究三维和四维空间的形状，使用从代数中产生的新的非对易数学。

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.