Applications of Noncommutative Algebra to Low-Dimensional Topology and Geometry

首席研究员建议继续调查的拓扑结构和几何的低维流形使用不变量，产生于非交换代数和冯诺依曼代数。 这反映了基本群的高度非对易性。在以前的工作中，PI证明了这种类型的不变量给出了3-流形的Thurston范数的估计，阻碍了3-流形在圆上的扩张，阻碍了某些4-流形上辛结构的存在，给出了关于链接协调群结构的新信息，并阻碍了一个群是3-流形的基本群或具有正亏。 PI提出寻找新的有趣的非交换代数不变量，并将这些不变量应用于低维拓扑中的问题;例如，3-流形的同调配边（和链接协调），4-流形的辛结构，3-流形的接触结构的属，3-流形的有限覆盖的贝蒂数，以及3-流形的叶理深度。PI还提出要找到一个特殊的关系，她的不变量的3流形和Heegard Floer同调的3流形。拓扑学是研究空间的连续变化（通过拉伸或扭曲，但不撕裂）。 在这个项目中，PI将专注于在3维空间（我们生活的空间）和4维空间（3维空间沿着时间维度）上局部建模的空间。 这些空间分别被称为3维和4维流形。我们可以更好地理解这些空间的方法之一是通过它们的“基本群”。“基本群是与任何拓扑空间相关联的代数对象，它度量空间中的洞的数量。 它被定义为在一个点开始和结束的循环的集合。 我们可以在基本群中形式上增加循环如下。 如果A是一个循环，B是另一个循环，我们定义“A乘以B”，记为AB，是通过首先遍历A然后遍历B得到的循环。 循环的乘法和矩阵乘法一样是不可交换的。 也就是说，AB与BA不同，因为遍历A然后遍历B与遍历B然后遍历A不同。不幸的是，空间的基本群很难理解。 在这个项目中，PI将使用非交换代数技术来更好地理解基本群，从而更好地理解三维和四维流形本身。 例如，PI将使用的非交换技术之一涉及不再有限但具有无限数量的行和列的矩阵。

The principal investigator proposes to continue the investigation of topology and geometry of low-dimensional manifolds using invariants that arise from non-commutative algebra and von Neumann Algebras. These reflect the highly non-commutative nature of the fundamental group. In previous work, the PI showed that this type of invariant gives estimates for the Thurston norm of a 3-manifold, obstruct a 3-manifold fibering over the circle, obstruct the existence of a symplectic structure on certain 4-manifolds, give new information about the structure of the link concordance group, and obstruct a group being the fundamental group of a 3-manifold or having positive deficiency. The PI proposes to find new interesting non-commutative algebraic invariants and to apply these invariants to questions in low-dimensional topology; for example, homology cobordism of 3-manifolds (and link concordance), symplectic structures of 4-manifolds, genera of contact structures of 3-manifolds, Betti numbers of finite covers of 3-manifolds, and depth of foliations of 3-manifolds. The PI also proposes to find a specific relationship between her invariants of a three manifold and the Heegard Floer Homology of a 3-manifold.Topology is the study of the continuous change of space (by stretching or twisting but not tearing). In this project, the PI will focus on spaces that are locally modelled on 3-dimensional space (the space that we live in) and 4-dimensional space (3-dimensional space along with a time dimension). These are called 3 and 4-dimensional manifolds respectively.One of the ways that we can better understand these spaces is via their "fundamental group." The fundamental group isan algebraic object associated to any topological space which measures the number of holes in a the space. It is defined as the set of loops starting and ending at a point. We can formally multiply loops in the fundamental group as follows. If A is a loop and B is another loop, we define "A times B", denoted AB, to be the loop obtained by first traversing A and then traversing B. The multiplication of loops is non-commutative as in matrix multiplication. That is, AB is not the same as BA, since traversing A then B is not the same as traversing B then A. Unfortunately,the fundamental group of a space is quite difficult to understand. In this project, the PI will use noncommutative algebraic techniques to better understand the fundamental group and hence better understand the 3 and 4-dimensional manifolds themselves. For example, one of the non-commutative techniques the PI will use involves matrices which are no longer finite but have an infinite number of rows and columns.