Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?

Question

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!)

Answer 1

I prefer this approach which I believe is due to Grothendieck. (I haven't checked how this compares with the sources cited by Nick Kuhn.)

Let $(\mathbb{K},R,d)$ be $(\mathbb{R},\mathbb{Z}/2,1)$ or $(\mathbb{C},\mathbb{Z},2)$. Let $V$ be a $\mathbb{K}$-linear vector bundle over $X$. Over the associated projective bundle $PV$ we have a $\mathbb{K}$-linear tautological line bundle $T$ classified by a map $PV\to \mathbb{K}P^\infty$. I'll assume that we know that $H^*(\mathbb{K}P^\infty;R)=R[x]$ with $|x|=d$. Pulling back $x$ gives a class $x\in H^d(PV;R)$. Induction over the cells of $X$ shows that $H^*(PV;R)$ is a free module over $H^*(X;R)$ with basis $\{x^i\mid 0\leq i<\dim(V)\}$. Thus, there is a unique monic polynomial $f_V(t)\in H^*(X;R)[t]$ of degree $\dim(V)$ such that $H^*(PV;R)=H^*(X;R)[x]/f_V(x)$. The characteristic classes of $V$ are just the coefficients of $f_V(t)$ (possibly with an extra $\pm$-sign, according to conventions). The cofibre of the inclusion $PV\to P(V\oplus W)$ is the Thom space of a $\mathbb{K}$-linear vector bundle over $PW$, and it follows that $f_V(x)f_W(x)$ annihilates $H^*(P(V\oplus W);R)$, and thus that $f_{V\oplus W}(t)$ must be equal to $f_V(t)f_W(t)$. By comparing coefficients we get the standard formula for characteristic classes of $V\oplus W$.

Answer 2

There are many ways to prove these formulae uniformly, and, of course, you need to start from an appropriate definition of these classes. But as has already been suggested in the comments, pretty much any definition of the Chern classes can be adapted to give a definition of the Stiefel-Whitney classes, with proofs of their properties also in parallel. Certainly Husemuller's book does this.

When I have taught a course on fiber bundles, I give the proof the Whitney sum formula as presented in [P. Conner and E. Floyd, The relation of cobordism to K-theory, SLNM 28 (1966)]. It also works for any complex oriented theory too (and, if I remember right, it is the proof followed by Husemuller).

Of course, in the end, one learns that these classes for a bundle that is the sum of line bundles are the symmetric functions, and then the fact that the formula is true becomes evident using the splitting principle.

Answer 3

This is mainly a comment about your point (4), but I do not seem to have enough reputation to comment on Oscar's comment.

In fact, there is an $\mathbb{E}_{\infty}$ ring map $$\Sigma_+^{\infty}\mathrm{BU} \to H\mathbb{Z}[t^{\pm}],$$ or equivalently an infinite loop map $$\mathrm{BU} \to GL_1(H\mathbb{Z}[t^{\pm}]).$$ The claim that the latter map is a loop map recovers the formula for the total chern class of a direct sum of bundles. This $\mathbb{E}_\infty$ ring map is constructed in the many author paper "Algebraic cycles and infinite loop spaces." Allen Yuan once convinced me that the algebraic geometry in their paper can be replaced by a homotopy theoretic argument, but nothing is written about that.

As a warning, to make sense of this one needs to take care about the meaning of $H\mathbb{Z}[t^{\pm}]$. There are several different $\mathbb{E}_\infty$ ring structures on $H\mathbb{Z}[t^{\pm}]$ (which are all equivalent as $\mathbb{E}_2$ rings). The precise $\mathbb{E}_\infty$ ring structure you want here is the one constructed in the many author paper "Algebraic cycles and infinite loop spaces." Of course, it is only the $\mathbb{E}_2$ structure that matters for such an unstructured statement as you want in your question.

I believe that what Totaro proves is that there is no homotopy ring spectrum whose underlying additive spectrum is $sl_1(H\mathbb{Z}[t^{\pm}])$, which Segal hoped would exist but seems like a rather strange thing to ask for from a modern point of view.