Factorization homology and the topology of manifolds

因式分解同调和流形拓扑

• 批准号: 1207758
• 负责人: John Francis
• 金额: $ 12.94万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207758&HistoricalAwards=false
• 关键词: Factorization; homology; topology; manifolds

Factorization homology is a homology theory for topological n-manifolds, constructed as a topological analogue of the homology of a factorization algebra in the algebro-geometric sense of Beilinson and Drinfeld. The coefficient system for such a theory is provided by a choice of n-disk algebra, a structure arising from n-fold loop spaces, Lie algebras, and certain quantum field theories, among other sources. This project aims to further develop this relatively new theory, to both bring new techniques to bear on previously studied invariants and to find new manifold invariants. A particular focus is 3-manifold topology, where one goal is to express quantum knot and 3-manifold invariants, such as Reshetikhin-Turaev invariants, Vassiliev knot invariants, and Chern-Simons invariants, in a uniform way in terms of factorization homology, and a second goal is to construct new knot homology theories via factorization homology. Another focus is the relation of factorization homology to the surgery classification of higher-dimensional topological manifolds.This project lies in topology, which studies abstract notions of space, motivated by mathematical physics - an intersection which had led to a great deal of work and cross-pollination. The problems are concerned with what possible global geometry may exist in a theoretical model for physical space or space-time and how this global geometry can be detected. It is interesting problem to try to describe the global geometry of such a candidate model for space-time by local observations. This project is one approach to this problem. Namely, if one were allowed to make local observations of some kind at different points throughout space-time, then compare them, how could one then reconstruct the global geometry from these compatible collections of observations? Factorization homology, the focus of this project, provides one avenue toward addressing this question.

分解同调是拓扑n流形的一种同调理论，在Beilinson和Drinfeld的代数-几何意义上构造为分解代数同调的拓扑类比。这种理论的系数系统是通过选择n盘代数、n折环空间、李代数和某些量子场论以及其他来源提供的。本项目旨在进一步发展这一相对较新的理论，为以前研究的不变量带来新的技术，并找到新的流形不变量。特别关注3流形拓扑，其中一个目标是在分解同调中以统一的方式表达量子结和3流形不变量，如Reshetikhin-Turaev不变量，Vassiliev结不变量和chen - simons不变量，第二个目标是通过分解同调构建新的结同调理论。另一个重点是高维拓扑流形的分解同调与手术分类的关系。这个项目属于拓扑学，它研究空间的抽象概念，受到数学物理学的启发——一个导致大量工作和交叉授粉的交叉点。这些问题涉及物理空间或时空的理论模型中可能存在的全局几何形状以及如何检测这种全局几何形状。试图用局部观测来描述这样一个候选时空模型的全局几何是一个有趣的问题。这个项目是解决这个问题的一种方法。也就是说，如果一个人被允许在整个时空的不同点进行某种局部观测，然后比较它们，那么一个人如何从这些兼容的观测集合中重建全局几何？这个项目的重点是分解同源性，它为解决这个问题提供了一条途径。