Classifying spaces for proper actions and almost-flat manifolds

对空间进行分类以实现正确的操作和几乎平坦的流形

基本信息

批准号：
EP/N033787/1
负责人：
Nansen Petrosyan
金额：
$ 12.63万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN033787%2F1
关键词：
Classifying spaces proper actions almost

项目摘要

In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives. In the last twenty years, non-positively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of right-angled Artin groups, and to investigate Brown's conjecture.Our second objective is on almost-flat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.

在这项研究中，我们将结合几何群体理论，托管和几何形状的技术，以实现两个目标。在过去的二十年中，非物性弯曲的空间和群体一直处于几何群体理论和拓扑的最前沿。 I. Agol对Thurston的虚拟Haken猜想的突破性解决方案强调了它们的重要性，该解决方案使用D. Wise开发的非物质弯曲的立方体复合物的机制。同样，在过去的十年中，为许多（非物有性弯曲的）组类别验证了Baum-Connes和Farrell-Jones的猜想，通过其分类空间为代数K-理论的计算铺平了道路。这些猜想连接了许多不同的数学领域，并且在拓扑，分析和代数方面具有遥远的应用。因此，时间是调查此类组的（有限）属性的正确时间，并构建用于使用适合计算的几何特性进行适当作用的空间进行分类的模型。我们的第一个目标是为某些重要类别的群体（例如Coxeter群体和外部自动形态式的Artin群体）构建此类模型，以对某些重要类别的行为进行分类，并研究Brown的猜想。我们的第二个目标几乎是flat歧管。这些歧管是M. Gromov引入的平面流形的概括。它们自然出现在对具有负截面曲率的Riemannian歧管的研究中，并在研究具有均匀界限截面曲率的崩溃流形的研究中起着关键作用。我们将研究这些流形的特性特性，例如旋转结构和恢复主义在现代流形理论中起着不可或缺的作用。自旋结构在量子场理论和数学物理学中具有许多应用。特别是，在平滑定向的歧管上的旋转结构的存在使人们可以定义纺纱片和狄拉克操作员，可以将其视为拉普拉斯式的平方根。狄拉克操作员对于描述粒子物理学中的费米子的行为至关重要。它也是在Atiyah-Singer索引定理中引起的纯数学中的重要不变，Connes的非共同差异几何形状，Schrodinger-Lichnerowicz公式，Kostant的Cubic Dirac Operator和许多其他领域。我们建议研究几乎流动流形的方法是由几何/拓扑与群体理论之间的相互作用引起的。这在很大程度上是由于这些流形的拓扑结构完全由其基本群体归类。