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 $\pi\_{2k} (BU, \mathbb{Q}) \simeq {\mathbb {Q}}$.



Chern numbers of primitive classes in BU
We have that if $f\colon S^{2k}\to BU$ is an actual map of topological spaces (it is a little bit unclear from your formulation if you assume this) then $\langle c\_k,[f]\rangle>$ is a multiple of $(k-1)!$ and all multiple are possible. See for instance Husemoller: Fibre bundles, Cor 18.9.8, GTM 20, Springer Verlag.


