Positive Curvature and F_0-Spaces

从有理同伦理论的角度推测，具有正截面曲率的偶数维黎曼流形具有f_0空间的结构，即具有正欧拉特征的有理椭圆空间。一方面，我打算用等变有理同伦理论的方法研究等距环面作用下具有正/非负曲率的（理性椭圆型）黎曼流形。另一方面，我计划进一步推导f_0 -空间的性质，更具体地说，例如建立某些类别的f_0 -空间的Halperin猜想。我将在对称流形和其他特殊的流形如双商（其中几乎所有已知的正曲率流形的例子都可以找到）上研究这个猜想。

Conjecturally, from the viewpoint of Rational Homotopy Theory, even-dimensional Riemannian manifolds with positive sectional curvature have the structure of an F_0-space, i. e. a rationally elliptic space with positive Euler characteristic. On the one hand I intend to investigate (rationally elliptic) Riemannian manifolds with positive/non-negative curvature under the assumption of isometric torus actions via an approach by equivariant Rational Homotopy Theory. On the other hand I plan to derive further properties of F_0-spaces, more concretely, to establish for example the Halperin conjecture for certain classes of F_0-spaces. I shall study this conjecture on manifolds with symmetry and on further special classes of manifolds like biquotients (among which almost all the known examples of manifolds with positive curvature can be found).