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

TL;DR

This paper establishes conditions under which certain topological spaces, including suspensions of spheres and Grassmannians, exhibit hyperbolic behavior in their homotopy groups, using $K$-theory as a key tool.

Contribution

It introduces a $K$-theory criterion for $p$-hyperbolicity of suspensions and applies it to complex Grassmannians, expanding understanding of hyperbolic properties in algebraic topology.

Findings

$S^n igvee S^m$ is $Z/p^r$-hyperbolic for all primes $p$ and $r eq 0$ when $n,m geq 1$.

Suspensions of complex Grassmannians are $p$-hyperbolic for all odd primes $p$ when $n geq 3$ and $0<k<n$.

Various spaces containing $S^n igvee S^m$ as a $p$-local retract are $Z/p^r$-hyperbolic.

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 $n,m \geq 2$, 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 $\Sigma Gr_{k,n}$ is $p$-hyperbolic for all odd primes $p$ when $n \geq 3$ and $0<k<n$. We obtain similar results for some related spaces.