# On the $v_1$ periodicity of the Moore space

## TL;DR

This paper advances the understanding of the $v_1$-periodic component in the Adams spectral sequence for Moore spaces by refining existing algebraic methods and proposing a new conjecture to verify Mahowald's long-standing conjecture.

## Contribution

It improves upon Palmieri's approach by using the endomorphism ring of the Moore space, leading to a new conjecture related to Mahowald's conjecture.

## Key findings

- Progress in verifying Mahowald's conjecture for Moore spaces.
- Development of a new algebraic framework using $End(M)$.
- Formulation of a new conjecture to connect with Mahowald's formulation.

## Abstract

We present progress in trying to verify a long-standing conjecture by Mark Mahowald on the $v_{1}$-periodic component of the classical Adams spectral sequence for a Moore space $M$. The approach we follow was proposed by John Palmieri in his work on the stable category of $A$-comodules. We improve on Palmieri's work by working with the endomorphism ring of $M$ - $End(M)$ thus resolving some of the initial difficulties of his approach and formulating a conjecture of our own that would lead to Mahowald's formulation.