Can we mimic the construction of Chern classes using real or quaternionic bundles? If so, do we get anything interesting?
My question concerns the construction of Chern classes in Bott and Tu's Differential Forms in Algebraic Topoogy. For those who don't have the book, let me first briefly sketch it. I believe they're following Grothendieck's 1958 article on Chern classes.
First, we define the Euler class for a rank 2 real bundle. Fix such a bundle $\pi: E \rightarrow M$ and put a Hermitian metric on it. Given a trivialization $\psi_\alpha$ of the bundle, this allows us to to talk about radial coordinates $r_\alpha$ and angle coordinates $\theta_\alpha$ on the fibers. On overlaps $U_\alpha\cap U_\beta$, we can measure the difference in the angles $\theta_\alpha$ and $\theta_\beta$ by a function $\phi_{\alpha\beta}$ (defined up to multiples of $2\pi$). We can then find forms $\xi_\alpha$ on $U_\alpha$ such that $$\frac{1}{2\pi} d\phi_{\alpha\beta} = \xi_\alpha - \xi_\beta,$$ and one can show that the $d\xi_\gamma$ forms agree on overlaps and form a global closed $2$-form, which gives a cohomology class in $H^2(M)$. This is the Euler class. The Chern classes are then defined by taking the projectivization $P(E)$ of $E$ and doing some clever things with the exact sequence $$0\rightarrow S \rightarrow \pi^*E \rightarrow Q \rightarrow 0$$ on $P(E)$, where $S$ is the universal subbundle, and using the fact that for a complex line bundle, its Euler class and $c_1$ are defined to be the same.
So, in outline, the process is to define $c_1$ for a line bundle directly then bootstrap up to higher $c_i$ on a general bundle.
My question is, does the analogous construction for real bundles work, and if so, do we get anything interesting? Given a real line bundle $E$, we can define its Euler class directly (Bott and Tu sketch the construction for rank $n$ bundles in sections 11 and 12), and then presumably apply the same kind of bootstrapping procedure.
I'm not very familiar with this material, but I see at least two problems immediately. First, the Euler class of a line bundle seems a little goofy. Instead of an angle coordinate, now we just have $\pm 1$. (For a rank two bundle, we're really looking at the corresponding $S^1$ bundle, so for a rank $1$ bundle we're looking at $S^0$.) Second, Bott and Tu's construction uses the fact that the de Rham cohomology of complex projective space is a polynomial ring. But for $\mathbb RP^n$ most of the integral cohomology groups are torsion, so the de Rham cohomology is going to be really boring. But maybe this can be avoided by using cohomology with integer coefficients instead of de Rham cohomology. (Bott and Tu only need the Leray theorem to do the construction, it seems, and I think there's a more general version of that for cohomology with integer coefficients, but I would need to double check.) Another thought is to switch to mod 2 cohomology, where the cohomology ring of $\mathbb R P^n$ is nice again. I would also be interested in an answer for the quaternionic bundle case, but I haven't thought about that at all.
To summarize, the projective bundle construction seems like a general recipe for cooking up characteristic classes. Is it? If not, what is special about the Chern classes that makes it work only for them?
The short answer is that if you try this Grothendieckian construction for real bundles and use mod 2 coefficients (in order to make the cohomology ring of $\mathbb RP^n$ nice) you get the Stiefel-Whitney classes. See the first section of these notes for details.
The use of the Euler class in Bott and Tu's construction is somewhat of a red herring in terms of the real generalization and motivated by the fact that we know in advance (or rather, Grothendieck knew, from previous study of these classes) that the top Chern class should be the Euler class, so the first Chern class of a line bundle should be its Euler class. This allows the slick, ad hoc presentation Bott and Tu use. In the linked file, we just pull back from $\mathbb RP^\infty$ to treat the line bundle case, which is less exciting (because it apes the standard method of constructing characteristic classes by pulling back cohomology classes from Grassmannians) but has the advantage of being correct.