The order of the attaching class in the suspended quaternionic quasi- projective space (Q1056958)

Let \(Q_ n(H)\) denote the quaternionic quasi-projective n-space. Then \(Q_ n(H)\) is formed from \(Q_{n-1}(H)\) by attaching the cell \(e^{4n- 1}\) via a map \(T_ n\). In this paper the author determines the order \(o_{n,k}\) of the homotopy class of the suspension \(E^ kT_ n\) as follows: if n is even then \(o_{n,k}=2\cdot (2n-1)!\) and, if n is odd then \(o_{n,\infty}=(2n-1)!\). The proof proceeds by using the results of \textit{M. Mimura} and \textit{H. Toda} [J. Math. Kyoto Univ. 3, 251-273 (1964; Zbl 0129.154)] on \(\pi_{4n-2}(Sp(n-1))\) and by invoking the K- theoretic consideration in order to evaluate the lower bound of \(o_{n,k}\).