Obstructions to positive curvature and symmetry

AbstractAward：DMS 1404670，首席研究员：Lee肯纳德，Guofang Wei流形的曲率是试图量化空间可能弯曲的方式，从欧几里得平面开始，其平坦性通过声明平面具有零曲率来捕获。 三维空间中的二维圆球具有正曲率，其大小反映了这样一种感觉，即当我们穿过一个大半径的球体时，它的切平面在空间中的转动比半径较小的球体的切平面慢：半径为R的球体具有曲率1/R^2。 曲率大于或等于零的流形多年来一直是一个活跃的研究领域，大多数已知的例子都是通过施加一定量的对称性产生的，其中球面及其大的旋转群是最对称的例子。这些研究项目将同伦理论工具引入到格罗夫和齐勒对非负曲率黎曼流形的研究中。 一些问题要研究遵循著名的定理和Adem，亚当斯，陈省身。 特别地，本文对单生成上同调环上的亚当斯定理进行了推广，并考虑了正曲流形基本群上的陈问题在对称性存在下的推广.

AbstractAward: DMS 1404670, Principal Investigator: Lee Kennard, Guofang WeiThe curvature of a manifold is an attempt to quantify the ways that a space may bend, beginning with the Euclidean plane, whose flatness is captured by declaring the plane to have zero curvature. The two-dimensional round sphere in three-dimensional space has positive curvature, of a size that reflects the sense that as one traverses a sphere of large radius, its tangent plane turns more slowly in space than the tangent plane to a sphere of smaller radius: the sphere of radius R has curvature 1/R^2. Manifolds of curvature greater than or equal to zero have been an active area of study for many years, with most known examples produced by imposing some amount of symmetry, with the sphere and its large group of rotations being the most symmetric example.These research projects bring homotopy-theoretic tools into the approach of Grove and Ziller to the study of Riemannian manifolds of non-negative curvature. Some of the questions to be studied follow on well-known theorems and conjectures of Adem, Adams, and Chern. In particular, a generalization is suggested of Adams' theorem on singly-generated cohomology rings and generalizations in the presence of symmetry for Chern's problem on the fundamental groups of positively curved manifolds are considered.