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

中文翻译：

---

局域球体和射影空间的同伦幂零性

为了$p$-局部球体$\mathbb {S}^{2m-1}_{(p)}$和$p >3$一个素数，我们证明同伦幂零满足$\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$, 关于任何同伦结合$H$- 结构上$\mathbb {S}^{2m-1}_{(p)}$. 我们也证明$\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$除了有限数量的素数之外的所有素数$p >3$. 然后，对于关联的循环空间$\mathbb {S}^{2m-1}_{(p)}$- 投影空间$\mathbb {S}^{2m-1}_{(p)}P(n-1)$， 和$m,n\ge 2$和$m\mid p-1$, 我们得出$\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.