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Are there any reference for the classification of orientable disk bundle over a closed surface? I am particularly interested in the case if the surface is $S^2,RP^2,T^2$ or the Klein bottle.

Many thanks!

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  • $\begingroup$ For linear disk bundles this can be found in theorem 3.4, Chapter 17 of Husemoller's "Fiber bundles", 3rd edition. Specifically, the Euler class gives a bijection between $[B, SO(2)]$ and $H^2(B)$ for any paracompact base $B$. For smooth or topological disk bundles you need to argue that the orientation-preserving diffeomorphism (of homeomorphism) group of the $2$-disk deformation retracts to $SO(2)$, and then again appeal to the above result. I don't have a reference handy for this fact about diff/homeo groups. $\endgroup$
    – Igor Belegradek
    Commented Jul 12, 2021 at 12:52
  • $\begingroup$ That the orientation-preserving diffeomorphisms of the 2-disk deformation retract onto $SO(2)$ is a consequence using isotopy extension of Smale's theorem about diffeomorphisms of $S^2$ in jstor.org/stable/2033664. $\endgroup$
    – skupers
    Commented Jul 12, 2021 at 14:09

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Fix a base space $B$. Taking boundaries gives an equivalence from the category of (isomorphisms of topological) disk bundles over $B$ to the category of (isomorphisms of topological) circle bundles over $B$. When $B$ is a surface the latter are also called “Seifert fibered spaces”. These are described in many different references. One very nice exposition is given by Allen Hatcher in “Notes on basic three-manifold topology” - you can download this from his webpage.

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  • $\begingroup$ The second sentence needs a reference. $\endgroup$
    – Igor Belegradek
    Commented Jul 12, 2021 at 14:32
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    $\begingroup$ A quick Google search throws up the Wikipedia page - en.wikipedia.org/wiki/Sphere_bundle - They simply say “Alexander trick”. Ah - perhaps you are suggesting that I should restrict the arrows in my categories to be isomorphisms? That sounds better… $\endgroup$
    – Sam Nead
    Commented Jul 12, 2021 at 19:03
  • $\begingroup$ I’ve made this change. Thank you. $\endgroup$
    – Sam Nead
    Commented Jul 12, 2021 at 19:12
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    $\begingroup$ The classification of circle bundles is contained in the classification of Seifert fibrations but the latter is more complicated than the former. $\endgroup$
    – Moishe Kohan
    Commented Jul 13, 2021 at 2:14

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