Classifying spaces for proper actions and almost-flat manifolds

在这项研究中，我们将结合几何群论、拓扑学和几何学的技术来努力实现两个目标。在过去的二十年里，非正曲线空间和群一直处于几何群论和拓扑学的前沿。它们的重要性被I.Agol用D.Wise开发的非正曲线立方体复形机制突破性地解决了瑟斯顿的虚拟Haken猜想所强调。此外，在过去的十年中，Baum-Connes猜想和Farrell-Jones猜想已经在许多(非正曲线)群类上得到了验证，为通过它们的分类空间进行代数K-理论和L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，并在拓扑学、分析和代数中有广泛的应用。因此，现在是研究这类群的(有限性)性质并为具有适合计算的几何性质的适当动作的空间构建模型的时候了。我们的第一个目标是为一些重要的群类，如Coxeter群和直角Artin群的外自同构群，构造适当作用空间的分类模型，并研究Brown猜想。第二个目标是在几乎平坦的流形上。这些流形是M.Gromov引入的平坦流形的推广。它们自然地出现在具有负截面曲率的黎曼流形的研究中，并且在具有一致有界截面曲率的折叠流形的研究中起着关键的作用。我们将要研究的这些流形的特征性质，如自旋结构和协边线，在现代流形理论中起着不可或缺的作用。自旋结构在量子场论和数学物理中有着广泛的应用。特别是，光滑可定向流形上自旋结构的存在允许定义旋量场和狄拉克算符，该算符可以被认为是拉普拉斯的平方根。狄拉克算符在描述粒子物理中费米子的行为时是必不可少的。它也是纯数学中的一个重要不变量，它产生于Atiyah-Singer指数定理、Connes的非对易微分几何、薛定谔-李氏公式、Kostant的三次Dirac算子等许多领域。我们提出的研究几乎平坦流形的方法源于几何/拓扑学和群论的相互作用。这在很大程度上是因为这些流形的拓扑是完全按它们的基本群分类的。

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.