HOMOTOPY NILPOTENCY OF LOCALIZED SPHERES AND PROJECTIVE SPACES

For the p-localized sphere S-p(2m-1) with p > 3 a prime, we prove that the homotopy nilpotency satisfies nil S-(p)(2m-1) < infinity, with respect to any homotopy associative H-structure on S-(p)(2m-1) . We also prove that nil S-(p)(2m-1) = 1 for all but a finite number of primes p > 3. Then, for the loop space of the associated S-(p)(2m-1) -projective space S-(p)(2m-1) P(n - 1), with m, n >= 2 and m vertical bar p - 1, we derive that nil Omega(S-(p)(2m-1) P (n - 1)) <= 3.