Metastable homotopy groups of Sp(n) (Q1110867)

Continuing the work of his previous paper [Osaka J. Math. 23, 867-880 (1986; Zbl 0617.57023)], the author determines the 2-primary component of the metastable homotopy groups \(\pi_ i(Sp(n))\) of the symplectic groups for \(4n+10\leq i\leq 4n+15\). Using e-invariant methods and the relationship between the homotopy groups of Sp(n), Sp/Sp(n) and of the stunted quaternionic quasi-projective spaces \(Q^{\infty}_{n+1}\) in the metastable range, the computation is actually of the 2-primary components of the stable homotopy groups \(\pi^ s_ i(Q^{\infty}_{n+1})\) of \(Q^{\infty}_{n+1}\) in the corresponding range by investigating the properties of the attaching maps of their top cells. The flavor of the lengthy main theorem about the 2-primary components of the homotopy groups in the title is given by the following partial statements: \[ If\quad n\geq 3,\quad then\quad \pi_{4n+15}(Sp(n))\cong Z/2+Z/2+Z/2,\quad if\quad n\equiv 0(4),\quad \cong Z/2+Z/2,\quad otherwise; \] \[ if\quad n\geq 3,\quad then\quad \pi_{4n+14}(Sp(n))\cong Z/2+Z/2+Z/2^{\alpha},\quad if\quad n\equiv 0(4),\quad \cong Z/2+Z/2^{\alpha}\quad otherwise; \] where in the last result the exponents \(\alpha\) are defined in terms of the James numbers.