一个改进的四周期定理和对称性下的一个Hopf猜想
An improved four-periodicity theorem and a conjecture of Hopf with symmetry
摘要：
20世纪30年代，H.霍普夫证明了正截面曲率的偶数维闭流形具有正的欧拉示性数。我们证明了这一点下的等距$T^4$ -行动的流形上的额外的假设，从以前的肯纳德，Wiemeler和威尔金定理改进假设一个$T^5$ -行动。更具体地说，这是通过给出一个合理的上同调分类可能的不动点组件。主要的新工具是一个改进的四周期性定理最初开发的Kennard通过使用示性类理论。作为第二个应用，我们给出了一个合理的上同调分类的封闭正曲偶数维流形没有奇有理上同调，承认等距$T^6$作用。
英文摘要：
In the 1930s, H. Hopf conjectured that a closed, even-dimensional manifold of positive sectional curvature has positive Euler characteristic. We show this under the additional assumption of an isometric $T^4$-action on the manifold, improving from previous theorems of Kennard, Wiemeler and Wilking assuming a $T^5$-action. More specifically, this is achieved by giving a rational cohomology classification of possible fixed point components. The main new tool is an improvement on the four-periodicity theorem originally developed by Kennard through the use of characteristic class theory. As a second application we give a rational cohomology classification of closed positively curved even-dimensional manifolds without odd rational cohomology that admit an isometric $T^6$-action.
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