$J$-homomorphism and homotopy

Question

We have Bott periodicity theorem for unitary group $U(n)$: $$ \pi_{i-1}^{s}(U) = \pi_{i-1}(U(m)) \simeq \pi_{i}(Gr_m(\mathbb{C}^{2m})) \simeq \pi_{i+1}(SU(2m)) \simeq \pi_{i+1}^{s}(U) .$$ So we can define a complex $J$-homomorphism $$ J:\pi_*(U) \rightarrow \pi^s_* .$$ Can we use complex $J$-homomorphism in order to cauculate some stable (or instable) groups of spheres?