Orientable surface bundles over the circle and their structure group

Question

I want to understand whether orientable surface bundles over the circle, i.e. with orientable total space, are always trivial, so I though I would revive an old post and ask for a few clarifications, but don't have enough credits, so I'll post them here instead.

1. What restrictions are placed on the structure group by orientability of the total sapce? (Also for more general fiber bundles.)

2. Is the argument (and affirmative answer) given to the old post the same for general fiber bundles over $\mathbb{S}^1$, as opposed to vector bundles?

I'll outline Ma Ming's answer (which I didn't fully understand) here for self-containedness:

Let $E \rightarrow \mathbb{S}^1$ be an $SL(n)$ (vector) bundle. Its classification depends on the homotopy class of $\mathbb{S}^0 \rightarrow SL(n)$ which is trivial, so $E$ is trivial.

Thanks!

Answer 1

No; in fact there are many interesting nontrivial such bundles. Here are some details.

Surface bundles over a circle $S^1$ with fiber an orientable surface $\Sigma_g$ can be constructed as follows. Let $f : \Sigma_g \to \Sigma_g$ be a diffeomorphism. The quotient of the product $\Sigma_g \times [0, 1]$ by the equivalence relation

$$(x, 0) \sim (f(x), 1)$$

defines a $3$-manifold called the mapping torus $M_f$ of $f$, with a map to $S^1$ coming from projecting to the second coordinate. All fiber bundles over $S^1$ arise in this way. I believe it is moreover the case that the diffeomorphism class of the total space depends only on the class of $f$ in the mapping class group $\text{MCG}(\Sigma_g) \cong \pi_0 \text{Diff}(\Sigma_g)$ and that $M_f$ is orientable iff $f$ is orientation-preserving, hence iff its image in the mapping class group lies in the orientation-preserving subgroup $\text{MCG}^{+}(\Sigma_g)$ of the mapping class group.

The easiest nontrivial case to understand here is $g = 1$, where we get torus bundles. Here the orientation-preserving mapping class group is

$$\text{MCG}^{+}(\Sigma_1) \cong \text{SL}_2(\mathbb{Z})$$

so any non-identity element of $\text{SL}_2(\mathbb{Z})$ gives rise to a nontrivial orientable $\Sigma_1$-bundle over $S^1$. You get $3$-manifolds exhibiting three of the eight Thurston geometries this way. For $g \ge 2$ see Nielsen-Thurston classification.