On homotopy nilpotency of the octonian plane $\mathbb{O}P^2$

Abstract

Let $\mathbb{O}P^2_{(p)}$ be the $p$-localization of the Cayley projective plane $\mathbb{O}P^2$ for a prime $p$ or $p=0$. We show that the homotopy nilpotency class $\textrm{nil} \Omega(\mathbb{O}P^2_{(p)})<\infty $ for $p>2$ and $\textrm{nil} \Omega (\mathbb{O}P^2_{(p)})=1$ for $p>5$ or $p=0$. The homotopy nilpotency of remaining Rosenfeld projective planes are discussed as well.