On the homotopy group \(\pi_{8n+4}(Sp(n))\) and the Hopf invariant (Q808037)

Let Sp(n) be the n-th symplectic group. Let \(Q^{\infty}_{n+1}\) be the stunted quaternionic quasi-projective space. For \(\pi_ k(Sp(n))\), the range \(4n+2\leq k\leq 8n+3\) is called the meta-stable range and the range \(k\leq 4n+1\) the stable range. When \(k\leq 8n+3\), the group \(\pi_ k(Sp(n))\) has been studied by various authors. The author investigates \(\pi_ k(Sp(n))\) of the first dimension beyond the metastable range. His main method is the so-called EHP-sequence. One of the results is stated as follows: Let \(n\geq 1\) and \(n+1\neq 2^ t\). Then, \(\pi_{8n+4}(Sp(n))\cong {\mathbb{Z}}/2\oplus {\mathbb{Z}}/2\oplus \pi^ s_{8n+5}(Q^{\infty}_{n+1}),\) where the first \({\mathbb{Z}}/2\) summand is generated by \(\mu_{8n+1}\) on \(S^ 3=Sp(1)\) and the second is generated by the Samelson square of the generator \(\sigma_ n\in \pi_{4n+2}(Sp(n))\), which is known to be a cyclic group. Similarly, the complex case, that is about U(n), is also studied.