$p$-hyperbolicity of homotopy groups via $K$-theory

Abstract

We show that $S^n\vee S^m$ is $\mathbb{Z}/p^r$-hyperbolic for all primes $p$ and all $r\in {\mathbb{Z}}^{+}$, provided $n,m\ge 2$, and consequently that various spaces containing $S^n\vee S^m$ as a $p$-local retract are $\mathbb{Z}/p^r$-hyperbolic. We then give a $K$-theory criterion for a suspension $\Sigma X$ to be $p$-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $\Sigma G{r}_{k,n}$ is $p$-hyperbolic for all odd primes $p$ when $n\ge 3$ and $0<k<n$. We obtain similar results for some related spaces.

arXiv comment30 pages, updated to match accepted version. To appear in Math. Z