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!



Classification of disk bundle over surfaces
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.


