Homotopy nilpotency of localized spheres and projective spaces

Abstract For the $p$ -localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$ , respect to any associative $H$ -structure on . We also {S}^{2m-1}_{(p)}= 1$ for all but finite number of primes Then, loop space associated -projective {S}^{2m-1}_{(p)}P(n-1)$ $m,n\ge 2$ and $m\mid p-1$ derive \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$