A combinatorial approach to the exponents of Moore spaces

TL;DR

This paper introduces a combinatorial method to analyze the exponents of Moore spaces, demonstrating that certain power maps are null homotopic under specific conditions, thereby strengthening classical results.

Contribution

It provides a new combinatorial approach to understanding the exponents of Moore spaces and extends classical results on the null homotopy of power maps.

Findings

Projection of the $p^{r+1}$-th power map is null homotopic for specified conditions.

Strengthens classical results on the exponent of the loop space of Moore spaces.

Applicable for odd-dimensional mod $p^r$ Moore spaces with $p>3$, $r>1$, and $n>1$.

Abstract

In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic piece containing the bottom cell $T^{2n+1}\{p^r\}$ is null homotopic for $n>1$, $p>3$ and $r>1$. This result strengthens the classical result that $\Omega T^{2n+1}\{p^r\}$ has an exponent $p^{r+1}$.