Classification of sphere bundles and vector bundles over a surface

algebraic-topologycharacteristic-classesfiber-bundlesgeometric-topologyvector-bundles

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?

Best Answer

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.

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