Orientable surface bundles over the circle and their structure group 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.


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

*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!
 A: 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. 
