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

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 $1\le n\le \infty$.

20 pages