I am reading the proof of Gauss-Bonnet Theorem from the Characteristic Classes by Milnor and Stasheff.
There is a statement in the proof which says that:
"But the only characteristic class in $H^2(\; ; \mathbb{C})$ for complex line bundles $\zeta$ is the Chern class $c_1({\zeta})=e(\zeta_{\mathbb{R}})$ (and its multiples)."
Can anyone help me prove that why the only characteristic class in $H^2(\; ; \mathbb{C})$ for complex line bundles $\zeta$ is the Chern class $c_1({\zeta})$ (or a reference from the same book would be also useful).
I am familiar with the sections on the Euler class and the Chern class from the same book. But my knowledge of this area is limited. Thankyou.
Edit: I think I got the half of it. Since my manifold is closed, oriented and 2-dimensional, I have $H^2(M;\mathbb{C}) \cong H_0(M;\mathbb{C})$. If $M$ is assumed to be connected (which is not in the statement of the theorem), then $H^2(M;\mathbb{C}) \cong \mathbb{C}$. So I think it would be enough to show that $c_1(\zeta)=e(\zeta_\mathbb{R})$ is non-zero.
I believe the following argument is due to Serre. A characteristic class taking values in $H^k(-, R)$ is, more or less by definition, a natural transformation from the functor $[-, BG]$ of isomorphism classes of $G$-bundles to the functor $H^k(-, R)$. By the Yoneda lemma applied to, say, the weak homotopy category (to my mind this is one of the most spectacular applications of the Yoneda lemma), such characteristic classes can be identified with classes in $H^k(BG, R)$. These are the universal characteristic classes of $G$-bundles, and characteristic classes of $G$-bundles are given by pulling them back along classifying maps $X \to BG$.
Specialized to complex line bundles $G = U(1)$ (or $G = \mathbb{C}^{\times}$), the corresponding classifying space is $BG = \mathbb{CP}^{\infty}$, and we can compute that its integral cohomology is a polynomial ring $\mathbb{Z}[c_1]$ on the universal first integral Chern class $c_1 \in H^2(\mathbb{CP}^{\infty}, \mathbb{Z})$. In particular, by universal coefficients this gives that $H^2(\mathbb{CP}^{\infty}, \mathbb{C})$ is $1$-dimensional and spanned by $c_1$.
The argument you give in your edit does not suffice; a characteristic class is a function from $G$-bundles to cohomology classes and two such functions may a priori fail to be linearly independent even though the cohomology group they take values in is $1$-dimensional. What we need is to compute the corresponding cohomology group of $BG$ as in the above argument.