Calculation of Chern number of $U(N)$ principal bundle I am considering $U(N)$ principal bundle on a two-dimensional sphere $S^2$. 
(Below the fiber is $N\times N$ unitary matrices for simplicity, and any generalization to general representations will be appreciated.) 
The bundle is defined by the transition function $t(\theta)\in U(N)$ along the equator $S^1$ parametrized by $\theta\in[0,2\pi)$. I am trying to calculate the first Chern number of this bundle and wondering whether it is generally possible to do a (globally defined, of course) gauge transformation to get an equivalent principal bundle defined by $\tilde{t}(\theta)$ so that
\begin{eqnarray}
\tilde{t}(\theta)\in \left[U(1)\right]^N, 
\end{eqnarray}
where $\left[U(1)\right]^N$ is the Cartan subgroup of $U(N)$, namely diagonal matrices. 
Does such gauge transformation exist generally? If so, the first Chern number can be always calculated by abelian subgroup. 
 A: Yes. And you can do better.
Observe that the clutching function  is determined by its homotopy class $t\in \pi_1(U(N))$. Note that there is a homeomorphism $U(N)\cong U(1)\times SU(N)$. Since $SU(N)$ is simply connected we get $\pi_1(U(N))\cong \pi_1(U(1))$, so we can choose $t$ to factor through the inclusion $U(1)\hookrightarrow U(N)$ up to homotopy as $t:S^1\xrightarrow{\bar t}U(1)\hookrightarrow U(N)$, and $U(1)$ certainly lies inside the diagonal subgroup $U(1)^N$.
So yes, the first Chern class of any bundle over $S^2$ (in fact over any 2- or 3-dimensional CW complex) is determined by the abelian subgroup $U(1)$. For higher dimensional spaces this will not always be true.
Note that this is not the same as the diagonal $U(1)$-subgroup, consisting of matrices $\lambda\cdot I_n$ with $\lambda \in S^1$. The inclusion of this subgroup induces multiplication by $N$ on $\pi_1$, since it factors as $\pi_1(U(1))\xrightarrow{\Delta}\pi_1(U(1)^N)\hookrightarrow \pi_1(U(N))$ where the first map is induced by the diagonal. This means that $t$ is deformable to a map in the image of this incusion if and only if it has degree $0\mod N$.
