Topological and equivariant rigidity in the presence of lower curvature bounds

作用在流形上的紧群理论有很长的传统，现在已经很好地理解了。群作用于度量空间和/或奇异空间的更一般情况目前主要是在非正曲线度量空间的背景下进行的。然而，现代几何的其他部分的基础是Alexandrov空间(曲率在下面有界)，它配备了紧致李群G的等距作用。这些度量空间包括，例如，具有较低截面曲率界限和等距G-作用的黎曼流形。黎曼几何学者特别感兴趣的是具有等距紧李群作用的具有正或非负曲率的黎曼流形。这种兴趣在很大程度上是由所谓的Grove程序推动的，该程序建议将具有正截面曲率或非负截面曲率的黎曼流形分类为一个大的等距群。人们可以考虑作用于流形的最简单的群是正维的紧致阿贝尔李群，即环面。到目前为止，光滑环面作用理论已经发展得很好了。在黎曼几何的背景下，特别是在Grove程序的背景下，具有环面作用的正弯曲黎曼流形已被广泛研究。由于几位作者的工作，最著名的是Grove和Searle，Fang和Rong，以及Wilking，只要流形或作用于流形的环面具有足够大的维度，就可以得到相当完整的分类结果。本文以光滑流形上光滑环面作用的Grove程序和成熟的上同调方法理论为出发点，一方面在具有较低截面曲率下界的黎曼流形上应用和发展等变拓扑方法，另一方面研究同调闭的Alexandrov闭空间。主要目标分别是得到具有有效等距2-环面作用的闭的、单连通的6-流形的拓扑分类和等变分类，以及分类同调的闭的、正弯曲的Alexandrov空间。这些问题的解决需要解决许多本身感兴趣的自含式问题，包括考虑具有环面作用的奇异空间(例如，奥比诺德空间和亚历山大空间)。

The theory of compact groups acting on manifolds has a long tradition and is quite well understood nowadays. The more general case of groups acting on metric and/or singular spaces is today mostly pursued in the context of non-positively curved metric spaces. However, fundamental to other parts of modern geometry are Alexandrov spaces (of curvature bounded below) equipped with an isometric action of a compact Lie group G. These metric spaces include, for example, Riemannian manifolds with a lower sectional curvature bound and an isometric G-action. Of particular interest to Riemannian geometers have been Riemannian manifolds with positive or non-negative curvature equipped with isometric compact Lie group actions. This interest is in great part propelled by the so-called Grove program, which proposes to classify Riemannian manifolds of positive or non-negative sectional curvature with a large isometry group.The simplest groups that one may consider acting on a manifold are compact abelian Lie groups of positive dimension, i.e., tori. By now the theory of smooth torus actions is well developed. In the context of Riemannian geometry and, in particular, of the Grove program, positively curved Riemannian manifolds with torus actions have been extensively studied. Thanks to the work of several authors, most notably Grove and Searle, Fang and Rong, and Wilking, fairly complete classification results are available, provided that either the manifold, or the torus acting upon it, has sufficiently large dimension. For the action of a circle and a 2-torus in dimensions 5 and 6, respectively, the usual methods have so far failed to yield topological and equivariant classification results.Taking as departure point the Grove program and the well-developed theory of cohomological methods for smooth torus actions on smooth manifolds, the present proposal aims, on the one hand, at applying and developing equivariant topological methods in the context of Riemannian manifolds with a lower sectional curvature bound and, on the other hand, at studying closed Alexandrov spaces of cohomogeneity one. The primary goals are, respectively, to obtain a topological and equivariant classification of closed, simply connected 6-manifolds with an effective, isometric 2-torus action, and to classify closed, positively curved Alexandrov spaces of cohomogeneity one. The solution of these problems entails solving many self-contained problems which are of interest in their own right, including the consideration of singular spaces (e.g. orbifolds and Alexandrov spaces) with torus actions.