Factorization homology and the topology of manifolds

因式分解同调是n维拓扑流形的同调理论，是Beilinson和Drinfeld的代数几何意义下的因式分解代数的同调的拓扑类比。这种理论的系数系统是通过选择n盘代数、由n重循环空间产生的结构、李代数和某些量子场论以及其他来源来提供的。这个项目的目的是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。其中一个特别的焦点是三维流形拓扑，其中一个目标是用因式分解同调统一的方式表示量子纽结和三维流形不变量，如Reshetikhin-Turaev不变量、Vassiliev纽结不变量和Chern-Simons不变量，第二个目标是通过因子分解同调来构造新的纽结同调理论。另一个焦点是分解同调与高维拓扑流形的外科分类的关系。这个项目位于拓扑学，它研究空间的抽象概念，受数学物理的启发-这一交叉导致了大量的工作和交叉授粉。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以及如何检测这种全局几何。试图通过局部观测来描述这样一个候选时空模型的全局几何形状是一个有趣的问题。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时间点进行某种局部观测，然后对它们进行比较，那么人们如何从这些兼容的观测集合中重建全局几何呢？因式分解同调是本项目的重点，它为解决这个问题提供了一条途径。

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.