Infinite families of higher torsion in the homotopy groups of Moore spaces

TL;DR

This paper refines the stable Snaith splitting for Moore spaces and constructs infinite families of higher torsion elements in their homotopy groups, revealing new structure in these groups.

Contribution

It introduces a refined splitting and constructs infinite $v_1$-periodic families of elements with higher torsion in Moore spaces' homotopy groups.

Findings

Infinite $v_1$-periodic families of elements of order $p^{r+1}$ are constructed.

Homotopy groups of the mod $p^{r+1}$ Moore spectrum are summands of unstable homotopy groups of Moore spaces.

Refinement of the stable Snaith splitting for double loop spaces of Moore spaces.

Abstract

We give a refinement of the stable Snaith splitting of the double loop space of a Moore space and use it to construct infinite $v_1$-periodic families of elements of order $p^{r+1}$ in the homotopy groups of mod $p^r$ Moore spaces. For odd primes $p$, our splitting implies that the homotopy groups of the mod $p^{r+1}$ Moore spectrum are summands of the unstable homotopy groups of each mod $p^r$ Moore space.