Cobordism in Low-Dimensional Topology

通过联合和个人的项目，利文斯顿，奥尔和Turaev调查光滑和拓扑一致性和协边性的结在三维，量子拓扑学和理论的虚拟结和字符串。配边的关系从根本上奠定了流形的拓扑结构，可以追溯到庞特里亚金，托姆，福克斯和米尔诺的开创性发现。该建议的主要目标是深化最近推出的几何和代数方法在三维结配边理论，并将这一理论扩展到更广泛的一类拓扑对象，包括虚拟结和虚拟字符串。我们的方法的新奇在于在经典的结理论，并在其扩展到cobordism和和谐的虚拟节点和链接的方法和技术的新组合。这将导致更好地理解结，链接，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-流形的基本群之间的相互作用。该研究为纽结理论和低维拓扑学方法在其他数学和理论物理领域的应用开辟了新的前景，包括Teichmuller理论、拓扑量子场论和组合词理论。经典纽结理论试图理解和分类三维空间中的闭合回路。 圆在空间中有多少种基本方式，我们如何在数学上区分这些方式？ 结模拟遗传结构和化学键。通过四维空间的纽结变形编码四维空间的基本基础，通过手术理论分类工具的四个流形。

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.