Some homotopy of the unitary groups detected by the \(K\)-theory of 2-cell complexes (Q1763032)

Let \(J\colon \pi_{4k-1}(SO(2m))\to \pi_{2m+4k-1}(S^{2m})\) the \(J\)-homomorphism and let \(x_{2m}\in \tilde{K}^0(S^{2m})\) be the Bott generator. Let \(k\geq 1\), \(m\geq 2k+1\), \(m\leq s\leq m+k\). Let \(j_{4k-1}\in \pi^S_{4k-1}\) the image under \(J\) of a generator of \(\pi_{4k-1}(SO)\). As \(\tilde{K}^0(S^{2m})\) is isomorphic to \([S^{2m},BU(s)]\cong {\mathbb Z}\), the composition \(x_{2m}\circ j_{4k-1}\) represents an element of \(\pi_{2m+4k-1}(BU(s))\). In the paper under review, the author proves that \(x_{2m}\circ j_{4k-1}\) is not equal to zero and its order is given by the denominator of \(\displaystyle{\frac{B_k}{4k}}\), if \(k\) is even and \(B_k\) is the \(k\)-th Bernoulli number. Partial results are also stated when \(k\) is odd.