3d bundles

Do you want a lot of really new problems in la topology? consider tri-dimensional fiber bundles $E$, of the form

$F\subset E\to B$

where by taking the exact combinations on the dimension of the fiber $F$ and the dimension of the base $B$,  to be $\dim F+\dim B = 3$, you will get many possibilities.

For example, if $F$ is a two-manifold (a surface) then you must choose $S^1$ to get non trivial surface bundles. Knowing that the mapping class group of the surface ${\cal{MCG}}(F)$,  classify the possible $E$‘s and since ${\cal{MCG}}(F)$ increases (depending which of three types of auto-homeomorphismus: periodic, reducible or pseudo-Anosov) its complexity as the genus of $F$ rises, then you will have a “bundle” of questions to tackle, to amuse, even to gain a PhD…