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 \(1\le n\le \infty \).