The Whitney sum formula for Stiefel-Whitney classes, $w_n(V \oplus W) = \sum w_i(V) w_{n-i}(W)$, looks a lot like the one for Chern classes $c_n(V \oplus W) = \sum c_i(V)c_{n-i}(W)$. But I don't know a way to prove both formulae "at once".

**Question:** Is there an abstract computation from which both Whitney sum formulae follow?

For instance, constructing cell structures on $BO$ and $BU$ and then just reading off what the relevant maps do on cohomology doesn't "explain" why the formulas "look the same" in both cases, so doesn't fit the bill for me.

**Notes:**

The Whitney sum formula for Stiefel-Whitney classes is also related to the Cartan formula for Steenrod squares, $Sq^n(xy) = \sum Sq^i(x)Sq^{n-i}(y)$, since Stiefel-Whitney classes can be defined in terms of Steenrod squares. I don't know a corresponding formula in integral cohomology operations related to Chern classes. An answer which sheds light on this connection would be most welcome.

These formulae are equivalent to working out the comultiplication on $H^\ast(BO;\mathbb F_2)$ and $H^\ast(BU;\mathbb Z)$ respectively. It appears to me to be forced algebraically that e.g. the pullback of $c_n \in H^\ast(BU(n);\mathbb Z)$ to $H^\ast(BU(1)^n;\mathbb Z)$ is some scalar multiple of $c_1^{\otimes n}$, but I can't seem to rule out that the scalar multiple is zero except by invoking the splitting principle (which I'd ideally like to avoid -- I'd really like a computation of this comultiplication which

*implies*the splitting principle!), and I'm also not sure I can show that if nonzero, the scalar is a unit.

**EDIT:** I did eventually arrive at a way to compute this coproduct without constructing complete cell structures: In the fibration $\mathbb{CP}^{n-1} \to BU(n-1) \times BU(1) \to BU(n)$, observe that the Serre spectral sequence for integral cohomology must collapse because we know the rank of all the groups involved, which are free. It follows that $H^\ast(BU(n)) \to H^\ast(BU(n-1) \times BU(1))$ is injective; iterating gives the splitting principle and the coproduct formula. The fibration $\mathbb{RP}^{n-1} \to BO(n-1)\times BO(1) \to BO(n)$ works similarly in the real case. I still find this argument unsatisfactory because one must still treat the two cases "in parallel" rather than proving one theorem and deducing both results from it.

There might be some sort of argument which deduces the comultiplication on $H^\ast(BO)$ from the one on $H^\ast(BU)$ or vice versa -- this isn't precisely what I'm looking for, but I'd be interested to see this worked out.

It would be very nice if there were an argument which were to abstractly construct an $E_\infty$ map $MU \to H\mathbb Z[t^\pm]$ (where $|t| = 2$) corresponding to the total Chern class $c = \sum_n c_n t^{-n}$, and deduce the comultiplication formula by computing that the map does this on the homology of $\Sigma^{\infty-1} BU \subset MU$ and using that it is multiplicative. I haven't quite been able to see through such an argument though. (As pointed out by Oscar Randal-Williams below, Totaro has shown that this does

*not*work!)