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

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 $1\le n\le \infty$.