On the Adams Filtration of a Generator of the Free Part of $π^s_*(Q_n)$

Let Qn be the quaternionic quasi-projective space (cf. [11]). Then it is a 2-connected CW-complex having one 4f— 1 dim cell for each i with l//in-i(On ; Z) is equal to a(n-l)-(2n-l}\ (see Theorem 3.10), where and throughout the paper a(i) denotes 1 if i is an even integer and 2 if i is an odd integer. Then, for any generator x of the free part of nln-i(Qn\ the modp Adams filtration Fp(x) is less than or equal to up(a(n— l)-(2n— 1)!), where vp(j) denotes the exponent of a prime p in the prime power decomposition of an integer /. Let GCp) denote the tensor product G®Z(P) for an abelian group G, where ZCp) is the ring of integers localized at p. Then, one of the results in this note is the following.