How to calculate characteristic classes of tensor products?

I was given the following as an execrise:

Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship:
$$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$

It is not clear to me how to solve this via the defining axioms of Chern classes. My personal guess is this should hold for other characteristic classes except the Pontrajin class (whose value on line bundles should be $0$). Since the problem should be simple, I decide to ask for a hint at here.

My thought on other approaches are as follows: given $E_{i}\rightarrow X$ we can form a classifying bundle $E’_{i}\rightarrow BU_{1}$ with a classifying map $X\rightarrow BU_{1}$ such that associated principal bundle of $E_{i}$ is the pull back of $E_{i}’$. Thus it is suffice to prove this for bundles over $BU_{1}$, but any line bundles should be the pull back of univeral line bundle over $BU_{1}$. So it suffice to prove this for universal line bundles over $BU_{1}$ and its pull backs to itself in general. However even this is not clear to me.

The way I recall we proved the Whitney sum formula is by making use the diagonal map $$B\iota:BU_{i}\times BU_{j}\rightarrow BU_{i+j}$$
where $U_{i},U_{j}$ are now placed on the diagonal blocks to $U_{i+j}$ and this map is the one inverted by taking the classifying space. However it is not clear to me how the map $$\chi:U_{1}\times U_{1}\rightarrow U_{1}: (z,w)\rightarrow zw$$
representing the tensor product works on the level of classifying spaces. Therefore I do not know how to describe the map $$B\chi: BU_{1}\times BU_{1}\rightarrow BU_{1}$$ at the level of cohomology rings. It seems to me that $B\chi^{*}$ is not injective in general.

I was given the hint that I should consider the isomorphism $$B(U_{1}\times U_{1})\cong BU_{1}\times BU_{1}$$ which I used implicitly in the proof above. The professor asked me to prove this myself and suggested using Kunneth formula afterwards. But still this is not immediate to me.

Solutions Collecting From Web of "How to calculate characteristic classes of tensor products?"

It would seem that Hatcher’s book on vector bundles and K-theory provides an answer at page 72. The proposition reads as follows:

The function $ w_1: Vect^1(X) \longrightarrow H^1(X;Z_2) $ is an isomorphism if $ X $ has the homotopy type of a CW complex. The same is also true for $ c_1 $ […]

As you suggested the proof treats the universal bundle case, and then uses pullback for the general case.

Here is an entirely diferent proof for Riemann surfaces.

Definitions:
Let $K_X$ be the canonical line bundle(cotangent bundle) of the riemann surface $X$, let $X_\alpha$ be a generic coordinate chart with holomorphic coordinate $z_\alpha: X \to \mathbb{C}$. And let your line bundle $E$ have transition functions $t_{\alpha,\beta}: X_\alpha \cap X_\beta \to \mathbb{C}$. Recall that a (hermetian)metric for $E$ is a positive section $h \in \Gamma(X, \bar{K_X} \otimes K_X)$ which in turn is just a collection of maps $h_\alpha: X_\alpha \to \mathbb{R}_{>0}$ such that $h_\alpha=|t_{\alpha, \beta}|^{-2}h_\beta$. Recall that the curvature of $E$ with respect to the metric,$F_E$, is (on the chart $X_\alpha$) $=-\partial_{\bar z_\alpha} \partial_{z_\alpha} log h$. And recall that the definition of the first chern number of $E$ is just $c_1(E)=\frac i {2\pi} \int_X F_E$.

Facts about the tensor products of line bundles that we prove:
Given line bundles $E_1$ and $E_2$ with transition functions on $X_\alpha \cap X_\beta$ given by $^{E_i}t_{\alpha, \beta}$, the transition functions for $E_1 \otimes E_2$ is just $^{E_1}t_{\alpha, \beta} \, ^{E_2}t_{\alpha, \beta}$.

As a result observe that if $E_1$ has a metric $h_1$ and $E_2$ has a metric $h_2$, that the collection of maps $h_{1, \alpha}h_{2,\alpha}:X \to \mathbb{R}_{>0}$ is a metric on $E_1 \otimes E_2$: This is because on$X_\alpha \cap X_\beta$, $h_{1,\alpha} h_{2, \alpha}= |^{E_1}t_{\alpha, \beta} \, ^{E_2}t_{\alpha, \beta}|^{-2}h_{1,\beta} h_{2, \beta}=|^{E_1\otimes E_2}t_{\alpha, \beta}|^{-2}h_{1,\beta} h_{2, \beta}$.

So the answer to your question is just that logarithm of a product is just the sum of logarithms and the metric on the tensor product of line bundles is just the product of the metrics: We have $c_1(E_1 \otimes E_2)=\int_X F_{E_1 \otimes E_2}=\int_X -\partial_{\bar z} \partial_{z} log h_1 h_2=\int_X -\partial_{\bar z} \partial_{z} log h_1 +\int_X -\partial_{\bar z} \partial_{z} log h_2=c_1(E_1)+c_1(E_2)$

This proof generalizes to a routine but tedious calculation to find the chern numbers of the tensor product of arbitrary vector bundles over a complex manifold(Exercise)(totally doable!).

If you interested in just case here is an another proof: As an exercise in undergraduate complex analysis and the definition above, one can show that $c_1(E)$=number of zeros of $\phi$ – number of poles of $\phi$, where $\phi$ is any meromorphic section of $E$. Here $\phi$ is regarded as a collection of maps from $\phi_\alpha:X_\alpha \to \mathbb{C}$ such that $\phi_\alpha=t_{\alpha ,\beta} \phi_\beta$ on $X_\alpha \cap X_\beta$. If $\phi$, $\psi$ are sections of $E_1$ and $E_2$ respectively, then the collection of maps $\phi_\alpha \psi_\alpha: X_\alpha \to \mathbb{C}$ defines a section of $E_1 \otimes E_2$, because $\phi_\alpha \psi_\alpha=^{E_1}t_{\alpha, \beta} \, ^{E_2}t_{\alpha, \beta}|^{-2}h_{1,\beta} \phi_\beta \psi_\beta=^{E_1\otimes E_2}t_{\alpha, \beta} \phi_\beta \psi_\beta$. Denote this section $\phi \psi$. Then $c_1(E_1 \otimes E_2)=zeros(\phi)-poles(\phi)+zeros(\psi)-poles(\psi)=zeros(\phi \psi)-poles (\phi \psi)=c_1(E_1)+c_1(E_2)$.

Final remark: I have given a formula for the first chern number $E$. You are interested in the first chern class in $H^2_{DR}(X)$. One definition of the first chern class of $E$ is just the class of the curvature tensor. Let me exaplain:

For this purpose observe that that the curvature tensor $F_E$(defined above and again below) is just a section of $K_X^2$. This is because the collection of maps $(F_E)_\alpha:=\partial_{\bar z_\alpha} \partial_{z_\alpha} log h_E$ from $X_\alpha \to \mathbb{C}$ can be viewed as a section of $K_X^2$ since $(F_E)_\alpha=(\partial_{z_\beta}/\partial_{z_\alpha})^2 (F_E)_\beta$ on $X_\alpha \cap X_\beta$. Since Riemann surfaces are of complex dimesion 1 and real dimension 2, $F_E$ is a 2-cocycle and one can easily verify that $[F_E] \in H^2_{DR}(X)$ satisfies the axioms that you need.

Denote the first chern class of $E$ by $C-1(E)$. Then we have $C_1(E_1 \otimes E_2)=[F_{E_1 \otimes E_2}]$. As above $F_{E_1 \otimes E_2}=F_{E_1}+F_{E_2}$ by the multiplicativity of transition functions and additivity of logarithms.