TITLE: Surface bundles over a surface
QUESTION [12 upvotes]: What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to
(1) homotopy?
(2) homeomorphism?
(3) fiberwise homeomorphism?
(4) bundle isomorphism?
And can these always be computed given 2 specific surface bundles over $\Sigma_h$?

REPLY [5 votes]: I'll make some comments, but I don't know the complete answer to your questions. (I'll assume $g,h>0$ too.)
As Thurston says, the bundle is determined up to isomorphism by the image $\pi_1(\Sigma_h)\to Mod(g)$. I think this answers questions (3) and (4). If two fiber bundles are fiberwise homeomorphic, then the space of fibers determines the base space, and therefore the fibration, so I think (3) and (4) are equivalent. The algorithmic question is open, as far as I know (of determining when the image of two surface groups in $Mod(g)$ are conjugate). 
I think questions (1) and (2) are more difficult. One issue is that a 4-manifold that fibers over a surface might fiber many different ways. Because I'm assuming $g,h>0$, these manifolds are $K(\pi,1)$'s, and so the homotopy question reduces to determining if $\pi_1$ are isomorphic. The homeomorphism problem would follow from the the Borel conjecture, but this is wide open in the case of 4-manifold with fundamental group of exponential growth.
There is a well-known open question, whether there is a convex-cocompact map (in particular injective) $\pi_1(\Sigma_h)\to Mod(g)$. If this exists, the $\pi_1$ of the associated bundle is a word-hyperbolic group. In this case, there is a theorem of Sela which allows one to algorithmically distinguish the fundamental groups, and therefore determine the homotopy type of the associated manifolds. However, this could be a theory of the empty set, since no examples are known!
Addendum: There are some techniques for distinguishing the diffeomorphism types of these manifolds. If the fiber $\Sigma_g$ is homologically non-trivial in the manifold, then Thurston proved that $M$ admits a symplectic structure. Work of Taubes implies that the Seiberg-Witten invariants of $M$ may be computed from the Gromov invariants of $M$. This might give methods for detecting when two bundles are not diffeomorphic. I suppose that such manifolds might have many different smooth structures, but it's not clear to me that the natural smooth structures associated to different fiberings could be different.
One might be able to distinguish homotopy types of these manifolds using group invariants. For example, these groups are subgroups of mapping class groups, and are therefore residually finite. One could count homomorphisms from the fundamental group to a finite group to obtain invariants that may distinguish the bundles.