Algebraic and Geometric Topology

关于高维流形的外科理论分类有一系列的猜想，在K-和l-理论中的Farrell-Jones猜想中达到顶峰。这些猜想与沿余维1子流形分裂流形密切相关。这个分裂问题又与代数K-和l -理论中的零群有关。这个项目有几个不同的方面。一是解决连通和问题——一个同伦等价于连通和的流形在什么情况下是连通和？另一个是加强了法雷尔-琼斯猜想在l理论中的作用，类似于戴维斯-汗-拉尼基在k理论中的作用。另一种是利用无限二面体群的l-理论计算和晶体群的l-理论中的Farrell-Jones猜想的证明，与Connolly合作，对环面上的对合进行分类。也有一些中低维问题，涉及透镜空间的自同伦等价的映射环面和连杆协调的某些方面。与几何拓扑中通常的目标一样，目标是使用各种代数、几何和分析技术来查找和计算分类的不变量。几何拓扑学是对流形的研究。一个n维流形是在n维欧几里德空间上局部建模的点的集合。例如，一个2流形是一个表面，在每个点附近看起来像一个平面。许多物理现象都是由流形表示的，因此，理解流形的整体结构，以及可能存在的流形，是科学和数学的基础。流形理论与数学的大多数领域，以及物理现象，如宇宙学、弦理论、经典力学和量子力学联系在一起。

There are a web of conjectures about the surgery theoretic classification of high-dimensional manifolds, culminating in the Farrell-Jones Conjectures in K- and L-theory. The conjectures are closely related to splitting manifolds along codimension one submanifolds. This splitting problem is in turn related to nil groups in algebraic K- and L-theory. This project has several different aspects. One is to solve the connected sum problem -- when is a manifold which is homotopy equivalent to a connected sum itself a connected sum? Another is to provide a strengthening of the Farrell- Jones Conjecture in L-theory, similar to that achieved by Davis-Khan- Ranicki in K-theory. Yet another is to classify involutions on a torus, using the computation of the L-theory of the infinite dihedral group and the proof of the Farrell-Jones Conjecture in L-theory for crystallographic groups, joint with Connolly. There are some middle and low dimensions problems too, involving mapping tori of self- homotopy equivalences of lens spaces and a certain aspect of link concordance.The goal, as usual in geometric topology, is to use a variety of algebraic, geometric, and analytic techniques to find and compute invariants for classification. Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Manifold theory connects with most areas of mathematics, as well as with physical phenomena such as cosmology, string theory, and classical and quantum mechanics.