How does one compute Chern numbers of spherical rational homology classes
$$f: S ^{2k} \to BU.$$ These generate rational homology by Milnor-Moore theorem since BU is a connected H-space, and so $c_k$ cannot kill such a class. It seems very likely that $\langle c_k,[f] \rangle =1$ but what is the proof?
Let me add here that I meant that $[f]$ is a Milnor-Moore generator, i.e. it's the multiplicative identity in $\pi_{2k} (BU, \mathbb{Q}) \simeq {\mathbb {Q}}$.