Some Problems in Curvature and Topology

摘要：DMS 0203164我们一直在研究黎曼几何中曲率与拓扑相互作用的问题，我们的主要工作是研究截面曲率以绝对值为界的坍缩流形的控制拓扑及其在全局黎曼几何中的应用。在过去的三年中，我们在Cheeger-Gromov-Fukaya开创的这一领域取得了实质性的进展：我们建立了高同伦群的同构有限的结果，建立了某类流形的微分同构有限的结果，以及该类中坍缩序列的收敛性。本文研究了非局部对称空间非正截面闭流形的一些刚性现象。我们在等距群包含大秩环面的正弯曲流形的同胚分类上取得了进展，并将这一结果推广到只允许等距离散阿贝尔群作用的更大的流形类。我们将在不久的将来继续我们在这些领域的项目，这代表了三年前为NSF拨款DMS 9971360提出的工作的延续。在过去的二十年里，黎曼几何最重要的发展之一是我们对黎曼流形的理解，这些流形看起来比它们的实际尺寸要小（即坍缩）。例如，一个非常薄的甜甜圈的表面看起来像一个圆，而它的曲率和直径是有限的。在曲率和直径有限的流形中，几乎所有的流形都比它们的实际尺寸小，这一惊人的事实使我们的进步得到了进一步的放大。

ABSTRACT: DMS 0203164.We have been studying problems in Riemannian geometry that concern with interplays between curvature and topology, and the major part of our work are about the controlled topology of collapsed manifolds with sectional curvature bounded in absolute value and applications to the global Riemannian geometry. For the past three years, we have made substantial progressesin this field pioneered by Cheeger-Gromov-Fukaya: we established the isomorphism finiteness result for the higher homotopy groups, the diffeomorphism finiteness result for a certain class of manifolds, and the convergence of collapsing sequences in this class. We have made a progress on investigating some rigidity phenomena for the class of closed manifolds of non-positive sectional which are not locally symmetric spaces. We have made a progress in the homeomorphism classification of positively curved manifolds whose isometry group contains a torus of large rank, and extend this result to the larger class of manifolds which only admit isometric discrete abelian group actions. We will continue our programs in these fields in the near future that represents a continuation of the work proposed three years ago for NSF Grant DMS 9971360. One of the most important developments in Riemannian geometry in the last two decades is our understanding of the Riemannian manifolds which appear to be smaller than their actual dimension (i.e collapsed). For instance, the surface of a very thin donuts looks like a circle while whose curvature and diameter are bounded. Our progress is amplified by the amazing fact that among the manifolds whose curvature and diameter are bounded, all but finitely many appear to be smaller than their actual dimension.