# Splitting principle and Chern roots

I have recently seen the following definition of Chern classes, relying on classifying spaces.

Let $$BG$$ be the classifying space of compact Lie group $$G$$. For the $$n$$-torus we have:

$$H^{**}(BT^n) = \mathbb Z[[x_1, \dots, x_n]],$$ where $$\{x_i\}$$ are the canonical basis of $$\mathrm{Hom}(T^n, U(1))\simeq H^2(BT^n)\simeq H^1(T^n)$$. (I use $$H^{**} = \prod H^k$$, which is larger than $$H^*(X)$$).

Now for $$U(n)$$ the Weyl group is the symmetric group $$S_n$$ and we can write: $$H^{**}(BU(n)) = \mathbb Z[[x_1, \dots, x_n]]^{S_n} = \mathbb Z[[c_1, \dots, c_n]],$$ where $$c_i$$ is the $$i$$th symmetric polynomial in $$\{x_k\}$$, hence $$c_i \in H^{2i}(BU(n)).$$

This construction is useful as it allows one to define classes via symmetric formal power series like $$\mathrm{ch} = \sum_k e^{x_k} \in H^{**}(BU(n); \mathbb Q).$$ Now if $$V\to X$$ is a complex vector bundle over a finite CW-complex X, we can define its chern class via $$\mathrm{ch}\,V := f^*\mathrm{ch} \in H^*(X; \mathbb Q),$$ where $$f\colon X\to EG$$ is the classifying map for the $$U(n)$$-bundle associated with $$V$$. (And I used the fact that $$X$$ is a finite CW-complex, so the Chern character is in $$H^*$$ – in general it should be in $$H^{**}$$).

Now I would like to prove that: $$\mathrm{ch}\,V \oplus W = \mathrm{ch}\, V + \mathrm{ch}\,W,\qquad (\star)$$ $$\mathrm{ch}\,V \otimes W = \mathrm{ch}\, V\smile\mathrm{ch}\,W \qquad (*)$$

Surely with the help of the splitting principle that essentially allows I to move the problem to another space $$Y$$ over which the pullback bundles $$V'$$ and $$W'$$ decompose into line bundles $$V' \simeq L_1 \oplus \dots \oplus L_n,\quad W' \simeq K_1\oplus \dots \oplus K_m.$$

Now I assume that I should move in some way from $$L_i$$ to $$x_i\in H^2(BT^n)$$ and from $$K_j$$ to $$y_j\in H^2(BT^m)$$, to get something like \begin{align*} \sum_i e^{x_i} + \sum_j e^{y_j} &= \sum_i e^{x_i} + \sum_j e^{y_i} \qquad&(\star')\\ \sum_i e^{x_i} \cdot \sum_j e^{y_j} &= \sum_{ij} e^{x_iy_j} \qquad&(*') \end{align*} My question is therefore:

How does one make this procedure rigorous?

I see two tricky points here:

1. How does one construct $$x_i$$ out $$L_i$$? (Moreover a single $$x_i$$ is not in $$H^2(BU(n); \mathbb Q)$$, so probably one constructs something symmetrized, as $$\sum e^{x_i}$$).
2. I have a strong feeling that the RHS of $$(\star')$$ should be expressed in terms of $$z_k \in H^2(T^{m+n}; \mathbb Z)$$ and there is some identification: $$x_i = \iota^*z_i$$, $$y_j=i^*z_{n+j}$$, where we have $$\iota\colon U(n)\times U(m)\hookrightarrow U(n+m)$$, which is the isomorphism of maximal tori $$T^n\times T^m \simeq T^{n+m}$$.

All ideas how to resolve these issues and suggestions of references will be very welcome.