Cobordism in Low-Dimensional Topology

通过合作和个人项目，Livingston， Orr和Turaev研究了三维结的光滑和拓扑一致性以及协同性，量子拓扑以及虚拟结和弦的理论。从庞特加金（Pontrjagin）、托姆（Thom）、福克斯（Fox）和米尔诺（Milnor）的开创性发现开始，协共关系从根本上奠定了流形拓扑的基础。该提案的主要目标是深化最近引入的几何和代数方法在三维结共体理论，并将该理论扩展到更广泛的拓扑对象类别，包括虚拟结和虚拟弦。我们方法的新颖之处在于在经典结理论中开发的方法和技术的新组合，以及它们对虚拟结和链接的协同性和一致性的扩展。这将有助于更好地理解结点、连杆、3-流形和4-流形的基本拓扑特征。主要研究人员期望更好地理解3流形和4流形基本群之间的相互作用。本研究将为结理论和低维拓扑方法在Teichmuller理论、拓扑量子场论和词的组合理论等数学和理论物理领域的应用开辟新的前景。经典结理论试图理解和分类三维空间中的闭环。圆在空间中有多少种基本形式，我们如何从数学上区分它们？结模拟遗传结构和化学键。结点在四维空间中的变形通过四流形的外科理论分类工具编码了四维空间的基本基础。

Through joint and individual projects, Livingston, Orr and Turaev investigate smooth and topological concordance and cobordism of knots in dimension three, quantum topology and the theory of virtual knots and strings. The relation of cobordism fundamentally underlies the topology of manifolds, dating from the seminal discoveries of Pontrjagin, Thom, Fox and Milnor. The principal goal of the proposal is to deepen the recently introduced geometric and algebraic methods in the theory of knot cobordism in dimension three and to extend this theory to a wider class of topological objects including virtual knots and virtual strings. The novelty of our approach lies in a new combination of methods and techniques developed in classical knot theory and in their extension to cobordism and concordance of virtual knots and links. This should lead to a better understanding of the fundamental topological features of knots, links, 3- manifolds, and 4-manifolds. Principal Investigators anticipate a better understanding of the interplay between the fundamental groups of 3-manifolds and 4- manifolds. The proposed study should open new perspectives for applications of methods of knot theory and low-dimensional topology in other areas of mathematics and theoretical physics, including Teichmuller theory, topological quantum field theory, and combinatorial theory of words.Classical knot theory attempts to understand and classify closed loops in three dimensional space. How many essential ways can circles sit in space, and how do we distinguish these mathematically? Knots model genetic structures and chemical bonds. Deformations of knots through four dimensional space encode the fundamental underpinnings of four dimensional space through the surgery theory classification tools for four manifolds.