Metastable Pseudoisotopy, G-Manifolds, and Functor Calculus

摘要奖：DMS 1608259，首席研究员：托马斯·G·古德威廉该奖项为主要研究员在代数拓扑学方面的研究提供支持，该研究人员使用代数工具来研究拓扑空间。其基本目标是找到将拓扑空间分类到一定等价性的代数不变量。流形是局部类似于欧几里德空间的拓扑空间。该项目的第一部分承诺为流形家族提供新的不变量，并在高维几何拓扑和弦拓扑之间建立新的联系。第二部分应该引出具有群作用的流形族的其他新的不变量。第三部分将从几何的角度继续研究“函子演算”，这是PI以前发展起来的一种理论。研究生将参与该项目并接受培训。计划中的研究包括三个部分。第一个涉及系统地检测外科手术和代数K-理论不可见的流形，从伪同余空间的Hatcher悬挂映射的同伦纤维开始。虽然K-理论的适当类比是难以捉摸的，但相应的拓扑环同调的类比和割圆迹似乎是可以达到的。第二个是关于有限群在流形上的作用；目标是发展与流形空间有关的各种结构的G版本，特别是h-余边空间的G版本。第三是关于同伦函子演算的“几何”观点，特别是关于在此背景下的喷射和微分算子的一些基本问题。

AbstractAward: DMS 1608259, Principal Investigator: Thomas G. GoodwillieThis award provides support for principal investigator's research in algebraic topology, which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to certain equivalence. Manifolds are topological spaces that locally resemble Euclidean spaces. The first part of the project holds the promise of new invariants for families of manifolds and new connections between higher-dimensional geometric topology and string topology. The second part should lead to other new invariants for families of manifolds with a group action. The third part will continue the study of "functor calculus", a theory developed by the PI previously, from a geometric point of view. Graduate students will be involved and trained in the project.The planned research has three components. The first involves systematically detecting that about manifolds which is invisible to surgery and algebraic K-theory, beginning with the homotopy fiber of the Hatcher suspension map for pseudoisotopy spaces. Though the appropriate analogue of K-theory is elusive, the corresponding analogue of topological cyclic homology and the cyclotomic trace seems to be within reach. The second is concerned with actions of finite groups on manifolds; the goal is to develop G versions of various constructions related to spaces of manifolds, and especially of spaces of h-cobordisms. The third is about the "geometric" view of homotopy functor calculus, and especially about some fundamental questions about jets and differential operators in that context.