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.


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