# Naive question: what good are characteristic classes of principal bundles?

I recently read a development of characteristic classes on principal bundles through curvature forms and the Chern–Weil homomorphism. Unfortunately, this exposition concluded without listing any specific applications. I've seen many applications of characteristic classes of vector bundles, and I know the associated bundle construction takes the former to the latter, but I don't yet know any specific applications of the principal bundle theory. So ...

What can one do with characteristic classes of principal bundles that can't be done with characteristic classes of vector bundles?

I'm certainly not an expert at this kind of problem, but, here is one example, which is found in K.Grove-W.Ziller, Lifting group actions and nonnegative curvature, Trans. Amer. Math. Soc. 363 (2011) 2865-2890. (And here is a link to a free version).

(Grove-Ziller) Suppose $E$ is a vector bundle over $\mathbb{C}P^2$ with non-vanishing second Steifel-Whitney class. Then $E$ admits a complete metric of non-negative curvature.

(They have similar results over all closed simply connected $4$-manifolds known to admit metrics of non-negative curvature: $S^4$, $\mathbb{C}P^2$, $S^2\times S^2$, and $\mathbb{C}P^2 \# \pm \mathbb{C}P^2$).

I know, you wanted an example using principal bundles, not vector bundles. This comes in the proof.

The idea is that every vector bundle can be written as an associated bundle $E = V\times_{O(n)} P$ for some principal $O(n)$ bundle $P$. (In fact, because each of the $M$s above is simply connected, one can use $SO(n)$ instead).

Since the usual metric on $V\cong \mathbb{R}^n$ is $O(n)$ invariant and non-negatively curved, the O'Neill formulas for a Riemannian submersion imply that if you can show $P$ has an $O(n)$ invariant metric of non-negative curvature, so does $E$.

So, how does one show that $P$ admits non-negative sectional curvature? Because $P$ is compact, it has a chance of being "built" out of compact Lie groups in nice ways - more specifically, $P$ has a chance of belonging to the class of compact homogeneous spaces, compact biquotients (quotients of homogeneous spaces by free isometric actions), or compact cohomogeneity one manifolds (spaces with a Lie group acting with codimension 1 orbits).

It turns out that many such $P$ are cohomogeneity one manifolds, which, by previous work of Grove and Ziller (K.Grove-W.Ziller, Curvature and Symmetry of Milnor Spheres, Annals of Mathematics 152 (2000),331-367.) admit metrics of non-negative curvature.

The characteristic classes enter in the following way. The Grove-Ziller construction allows them to construct many $SO(n)$ principal bundles $P\rightarrow M$, but a priori, it's not clear which bundles are actually constructable in this manner. However, their construction allows the to compute the characteristic classes of the tangent bundle to $P$, which in turn allows them to compute the characteristic classes of the principal bundle $P\rightarrow M$ itself. Finally, Dold and Whitney (A. Dold and H. Whitney, Classification of oriented sphere bundles over a 4-complex, Ann. of Math. (2) 69 (1959), 667-677.) have already shown that the characteristic classes of an $SO(n)$ principal bundle over a $4$ manifold completely classify the bundle. This allows Grove and Ziller to recognize that, for example, they have all $P\rightarrow \mathbb{C}P^2$ with non-vanishing second Stiefel-Whitney class.