Applications of Noncommutative Algebra to Low-Dimensional Topology and Geometry

首席研究员建议继续使用非交换代数和冯·诺伊曼代数产生的不变量来研究低维流形的拓扑和几何。这反映了基群的高度非交换性。在前人的工作中，PI证明了这类不变量给出了3流形的Thurston范数的估计，阻碍了3流形在圆上的纤维，阻碍了某些4流形上辛结构的存在，给出了链路协调群的结构的新信息，并阻碍了一个群是3流形的基群或有正缺陷。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问题；例如：3-流形的同调配合（和连杆协调）、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Betti数、3-流形的叶状深度等。PI还提出了寻找她的三流形不变量与三流形的Heegard flower同调之间的特定关系。拓扑学研究的是空间的连续变化（通过拉伸或扭曲而不是撕裂）。在这个项目中，PI将专注于局部以三维空间（我们生活的空间）和四维空间（三维空间和时间维度）为模型的空间。它们分别被称为三维流形和四维流形。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，它测量空间中孔的数量。它被定义为从一个点开始和结束的循环的集合。我们可以在基本群中正式地将环路相乘，如下所示。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，是先遍历A，再遍历B得到的循环。循环的乘法与矩阵乘法一样是不可交换的。也就是说，AB不等于BA，因为遍历A然后B不等于遍历B然后A。不幸的是，空间的基本群是很难理解的。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3维和4维流形本身。例如，PI将使用的一种非交换技术涉及的矩阵不再是有限的，而是具有无限数量的行和列。

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.