2
$\begingroup$

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!

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ 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
    Jul 12, 2021 at 14:09

1 Answer 1

5
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ The second sentence needs a reference. $\endgroup$ Jul 12, 2021 at 14:32
  • 1
    $\begingroup$ 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
    Jul 12, 2021 at 19:03
  • $\begingroup$ I’ve made this change. Thank you. $\endgroup$
    – Sam Nead
    Jul 12, 2021 at 19:12
  • 3
    $\begingroup$ The classification of circle bundles is contained in the classification of Seifert fibrations but the latter is more complicated than the former. $\endgroup$ Jul 13, 2021 at 2:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.