Topological and equivariant rigidity in the presence of lower curvature bounds

作用于流形的紧群理论有着悠久的传统，现在已经被很好地理解了。群作用于度量空间和/或奇异空间的更一般的情况，今天主要是在非正弯曲度量空间的背景下进行的。然而，现代几何其他部分的基础是亚历山德罗夫空间（曲率界如下），它具有紧化李群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正曲率或非负曲率的黎曼流形，它们具有等距紧致李群作用。这种兴趣在很大程度上是由所谓的格罗夫计划推动的，该计划提出用一个大的等长群对正或非负截面曲率的黎曼流形进行分类。可以考虑作用于流形上的最简单群是正维紧致阿贝尔李群，即环面。到目前为止，光滑环面作用理论已经得到了很好的发展。在黎曼几何，特别是格罗夫规划的背景下，具有环面作用的正弯曲黎曼流形得到了广泛的研究。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，就可以获得相当完整的分类结果。对于5维和6维的圆和2环面的作用，目前常用的方法都不能得到拓扑和等变的分类结果。本文以光滑流形上光滑环面作用的Grove规划和成熟的上同调方法理论为出发点，一方面应用和发展了具有低截面曲率界的riemann流形的等变拓扑方法，另一方面研究了同质性为1的闭合Alexandrov空间。主要目标分别是获得具有有效等距2环面的封闭单连通6流形的拓扑和等变分类，以及对同质性为1的封闭正弯曲Alexandrov空间进行分类。这些问题的解决需要解决许多自包含的问题，这些问题本身就很有趣，包括考虑具有环面作用的奇异空间（例如轨道和Alexandrov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.