Exponents of $[\Omega(\mathbb S^{r+1}), \Omega (Y)]$

TL;DR

This paper explores the exponents of certain homotopy groups related to loop spaces of spheres and other spaces, establishing connections with homotopy exponents of spheres and analyzing specific cases involving projective spaces.

Contribution

It determines the $p$-primary exponents of homotopy groups of loop spaces of spheres and extends the analysis to spaces like projective spaces and homotopy spheres.

Findings

$p$-primary exponents of $[\, ext{loop space of spheres} ext{, loop space of spheres} ext{]}$ match sphere homotopy exponents

Exponents for spaces with homotopy types of projective spaces are studied

Results connect exponents of loop spaces with classical homotopy exponents

Abstract

We investigate the exponents of the total Cohen groups $[\Omega(\mathbb S^{r+1}), \Omega(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n+1})]$ and $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $\Sigma(n)/G$ for a homotopy $n$-sphere $\Sigma(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$.