The Euler characteristic as an obstruction to compact Lie group actions

Abstract:

Actions of compact Lie groups on spaces $X$ with ${H^{\ast }}(X,{\mathbf {Q}}) \cong {\mathbf {Q}}[{x_1}, \ldots ,{x_n}]/{I_0}$, $Q \in {I_0}$ a definite quadratic form, $\deg {x_i} = 2$, are considered. It is shown that the existence of an effective action of a compact Lie group $G$ on such an $X$ implies $\chi (X) \equiv O(|WG|)$, where $\chi (X)$ is the Euler characteristic of $X$ and $|WG|$ means the order of the Weyl group of $G$. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree ${N_t}(G/T)$ is equal to $\dim G$ if $G$ is a compact connected Lie group and $T \subset G$ its maximal torus. Effective actions of compact connected Lie groups $K$ on $G/T$ with $\dim K = \dim G$ are completely classified.