Return to Revisions

You can identify $c_2$ with an element of $H^4(BU(2))$. Let $BU(2)\to K({\mathbb Z}, 4)$ be a map that classifies $c_2$. It induces a homomorphism $\pi_4(BU(2))\cong [S^4, BU(2)]_*\to [S^4, K({\mathbb Z}, 4)]_*\cong H^4(S^4)$. Your first question is equivalent to asking whether this homomorphism is an isomorphism. It follows from Bott's integrality theorem that the analogous homomorphism $\pi_{2n}(BU(n))\to H^{2n}(S^{2n})$ that classifies $c_n$ is multiplication by $(n-1)!$ (see for example Husemoller's book Fibre bundles, Corollary 9.8). In your case $n=2$, so it is an isomorphism.