Homotopy nilpotency of localized spheres and projective spaces

Abstract

Abstract 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 $\\mbox {nil}\\ \\Omega (\\mathbb {S}^{2m-1}_{(p)}P (n-1))\\le 3$.