On Samelson products in Sp(\(n\)) (Q839312)

Let \(Q_n\) be the quasi-projective space for \(Sp(n)\) which has the cell structure \(S^3 \cup e^{7} \cup \cdots \cup e^{4n-1}\). Then there exists natural inclusion map \(\epsilon_n: Q_n\to Sp (n)\). The Samelson product \(\langle \epsilon_n, \epsilon_n \rangle\) is defined by \(Q_n \wedge Q_n \overset{\epsilon_n \wedge \epsilon_n }{\rightarrow} Sp(n) \wedge Sp(n) \overset{\gamma}{\rightarrow} Sp(n)\) where \(\gamma\) is the commutator map. The fibre sequence \( \Omega Sp(\infty) \overset{\Omega \pi}{\rightarrow} \Omega(Sp(\infty)/Sp(n)) \overset{\delta}{\rightarrow} Sp(n) \overset{i}{\rightarrow} Sp(\infty)\) induces an exact sequence of groups \[ \widetilde{K Sp}^{-2}(X)\overset{(\Omega \pi)_*}{\longrightarrow} [X,\Omega (Sp(\infty)/Sp(n))] \overset{\delta_*}\longrightarrow [X, Sp(n)] \overset{i_*}\longrightarrow \widetilde{KSp}^{-1}(X). \] Since \(Sp(\infty)\) is homotopy commutative, \(i_*(\gamma)=0\), so there exists a lifting \(\tilde\gamma: Sp(n)\wedge Sp(n)\to \Omega (Sp(\infty)/Sp(n))\) of \(\gamma\) such that \(\delta\circ \tilde\gamma \simeq \gamma\). From this the author obtain the relation \(\langle \epsilon_n,\epsilon_n \rangle= \delta_*( \tilde\gamma\circ (\epsilon_n \wedge \epsilon_n ) )\). Using this relation the author proves that the order of \(\langle \epsilon_2, \epsilon_2\rangle\) is 280, and at odd primes the order of \(\langle \epsilon_n, \epsilon_n \rangle\) is \({3^4\cdot 5\cdot 7\cdot 11}\) for \(n=3\), \({3^5\cdot 5^2\cdot 7\cdot 11\cdot 13}\) for \(n=4\), and \({3^5\cdot 5^3\cdot 7^2\cdot 11\cdot 13 \cdot 17\cdot 19}\) for \(n=5\).