$\mathbb{Z}/p^r$-hyperbolicity via homology

TL;DR

This paper investigates the homotopy groups of Moore spaces and establishes conditions under which they are $ ext{Z}/p^r$-hyperbolic, providing a complete resolution for certain primes and exponents, and offering homology-based criteria.

Contribution

It determines when Moore spaces are $ ext{Z}/p^r$-hyperbolic for various primes and exponents, extending previous work and providing new homology criteria.

Findings

Homotopy groups of Moore spaces are $ ext{Z}/p^s$-hyperbolic for $s \,\leq r$ when $p^r eq 2$.

Complete characterization of $ ext{Z}/p^s$-hyperbolicity for Moore spaces for $p \,\geq 5$ and $p=2, r \,\geq 6$.

Homology criteria for $ ext{Z}/p^r$-hyperbolicity and examples.

Abstract

We show that the homotopy groups of a Moore space $P^n(p^r)$, where $p^r \neq 2$, are $\mathbb{Z}/p^s$-hyperbolic for $s \leq r$. Combined with work of Huang-Wu, Neisendorfer, and Theriault, this completely resolves the question of when such a Moore space is $\mathbb{Z}/p^s$-hyperbolic for $p \geq 5$, or when $p=2$ and $r \geq 6$. We also give a criterion in ordinary homology for a space to be $\mathbb{Z}/p^r$-hyperbolic, and deduce some examples.