# Chern numbers of primitive classes in BU

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

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.
• I read the proof and it is pretty clear except for one part in the proof of periodicity. He looks at the Bott map $\beta:\tilde K(X)\to \tilde K(X\wedge S^2)$ coming from $\xi\mapsto \xi\otimes(\gamma -1)\in \tilde K(X\times S^2)$ where $\gamma$ is the canonical complex line bundle over $S^2=\mathbb{CP}^1$. He claims that this map is obviously onto. I don't quite see why that's true. Nov 3, 2011 at 16:38