# Sphere fibrations over Riemann surfaces

In a physics application, there's a 5-dimensional space which consists of a fibration of $S^3$ over a genus-$g$ Riemann surface $\Sigma_g$. In fact this nontrivial fibration is part of a bigger where the remaining 5 dimensions are along an AdS$_5$ space. [Such a metric shows up in a string theory application -- I'll be happy to describe it if someone is interested.]

Forgetting for the moment the AdS$_5$ part, I would like to understand how to compute characteristic classes for this manifold.

In fact, what I actually want to compute is

$$\int_{\mathcal{M}} tr(R \wedge R)$$

(where $R$ is the curvature 2-form) and I want to do it without actually having to look at the details of the metric (I think these quantities are topological invariants, so there should be a cleverer way to compute them rather than using the physicist's standard way of solving Maurer-Cartan equations for spin-connection 1-forms, then computing the curvature 2-forms).

I am relatively new to algebraic topology but I have reason to believe that there are systematic ways to compute such quantities, so if anyone can point me to the right references in the math literature and/or give me a blurb, I'll be very grateful.

Thank you!

• I already do not understand the first sentence: Do you mean that you have some 5-dimensional manifold M which is the total space of a fiber bundle over $\Sigma_g$ with the fiber $S^3$? The 3-dimensional sphere itself is fibered over a surface, but this surface is $S^2$ and the fibers are circles; this is called the Hopf bundle. As for your question, yes, there are other ways to compute characteristic classes of fiber bundles without using a metric, but the integral you wrote is not how one computes char classes. – Moishe Kohan Apr 5 '17 at 22:56
• Thanks for your reply @MoisheCohen. I think I flipped the order around. Yes, that's what I mean. I will fix the order in a few hours when I get to my laptop. I know one doesn't use the integral to compute characteristic classes. As long as I am guaranteed the existence of a 4-cycle to integrate over (which I think the Gysin sequence tells me, am I right?) I can do the integral...and now, I'd like to know how to find its value. – leastaction Apr 8 '17 at 7:38
• Just to narrow down the setting of the problem, any smooth fibration over a Riemann surface with fibers $S^3$ is actually a locally trivial fiber bundle with fiber $S^3$ by Ehresmann's theorem. Furthermore, one can prove that any smooth locally trivial fiber bundle with fibers $S^3$ over a Riemann surface is smoothly bundle isomorphic to either the trivial bundle or just another (nontrivial) bundle. Since you are talking about a non-trivial $S^3$ fibration over a Rieman surface, that fibration is unique (up to isomorphism). So you are looking for only one case, so one computation is enough. – Futurologist May 15 '17 at 6:18
• Thus, you can think that you are given (the only one) $S^3$ non-trivial principal bundle over a Riemann surface. Is this curvature form $R$ the Riemann curvature of a Riemannian metric on it or is it the curvature of some connection? Is this metric $S^3$ invraint by any chance? – Futurologist May 15 '17 at 6:26