Classification of sphere bundles and vector bundles over a surface

Question

The general question for which I want an answer is:

Given $n\geq3$ and a closed surface $S$ of genus $g\geq1$, what are all the rank $n$ real vector bundles over $S$ (up to isomorphism)? What are all the $(n-1)$-sphere bundles over $S$?

Here "sphere bundle" means a fiber bundle whose fibers are spheres (with structure group the diffeomorphism group or homeomorphism group of the sphere). Let me point out what I know and where I need clarifications:

1. Are the two questions equivalent? The answer in this post seems saying that when $n\leq4$, they are, but this is not true for all $n$, and there is a distinction between the diffeomorphism and homeomorphism categories. Could someone give more details and references for this?

2. Restricting to the first question, I know that a necessary condition for two vector bundles to be isomorphic is to have the same characteristic classes. Since $n\geq3$ and $\dim S=2$, the only nontrivial characteristic classes in my situation are the 1st and 2nd Stiefel-Whitney classes. So the remaining question is: Is the condition also sufficient? Namely, are the rank $\geq3$ real vector bundles over a surface completely classified by the Stiefel-Whitney class?

Answer 1

The questions are not obviously equivalent, sphere bundles (whether smooth or topological) do not necessarily come from vector bundles. The first counterexample to this appears in dimension 4 thanks to the work of Watanabe.

As for the first question, it is easy to show that if you remove a disk from a surface, all orientable vector bundles over the surface are trivial. Hence, by picking a map $S^1 \rightarrow SO(n)$, we can obtain all bundles over a closed surface by gluing a trivial bundle over the disk to a trivial bundle over the punctured surface via the map.

We know $\pi_1(SO(n))$ is trivial if $n=1$, is $\mathbb{Z}$ if $n=2$, is $\mathbb{Z}/2$ if $n\geq 3$. In fact, one can use characteristic classes to see the assignment $\pi_1(SO(n)) \rightarrow \operatorname{Vect}_n(S_g)$ is injective as well. Notably, if $n=1$ this is trivial, if $n=2$ it is detected by the first Chern class, and if $n \geq 3$ it is detected by the second Stiefel-Whitney class.

So in dimensions $\leq 3$ we know the answer is the same for sphere bundles, but as soon as we get in higher dimensions it becomes much more difficult. The fundamental ingredient will still be $\pi_1$ of $\operatorname{Homeo}_+(S^n)$ or $\operatorname{Diff}_+(S^n)$. Perhaps it is already known if $n \neq 4$ that $\pi_1(SO(n)) \rightarrow \pi_1(\operatorname{Diff}_+(S^n)) \rightarrow \pi_1(\operatorname{Homeo}_+(S^n))$ is an isomorphism. In this case, the same argument would give the same classification.