Rational homotopy theory of flag manifolds

Abstract

Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $\pi _{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic Weyl invariants of $G$ to suitable simple factors of $P$, which also address a key nonvanishing issue not settled by earlier approaches of Meier, Shiga-Tezuka, Kotschick, and Terzic.