Surface bundles over a surface

Question

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 $\Sigma_h$?

Answer 1

I'm supposing you mean for $g, h > 0$. Associated with a surface bundle, there is a homomorphism of $\pi_1(\Sigma_h)$ to the outer automorphism group of $\pi_1(\Sigma_g)$. This is equivalent (with slight low-genus modifications) to a homotopy class of maps into the modular orbifold, Teichmüller space modulo the mapping class groups. The group of homeomorphisms of a surface homotopic to the identity is contractible, so these bundles are determined up isomorphism that acts as the identity on the base and on one fiber.

The conjugacy problem for the mapping class group is solved, using either the theory of pseudo-Anosov homeomophisms or automatic group theory, and either of those tools allows you to solve isomorphism up to bundle maps that are the identity on the base. Peter Brinkmann's program xtrain, which you can find online, computes the dilatation constant, which is typically enough to distinguish conjugacy classes in the mapping class group. Snappea, also available online, will usually distinguish homeomorphism classes of the 3-manifolds obtained by an element of the mapping class group (with exceptions that can be analyzed). This will also distinguish conjugacy classes, by looking for homoemorphism preserving a cohomology class.

The action of the mapping class group of the base on bundle maps seems trickier, and I don't think I know an immediate answer of classifying them. The troublesome cases would be where the image of the surface group in the mapping class group is not a quasi-isometric map of groups.

A classification of homeomorphism types would include the special case when the surface bundle is induced from a map of the base to a circle, so the bundle comes from a 3-manifold that fibers over a circle. 3-manifolds can fiber in many different ways, so not all homeomorphisms in these cases are fiber preserving, and the homeomorphism classification for these particular cases is solvable, but it gets into a complicated theory that won't usually work for 4-manifolds. I'm not sure what's known about surface fiber bundles over surfaces that fiber in multiple ways, apart from these.

One other point: the fundamental group of such a 4-manifold has an action on $S^1$, namely, the circle at infinity to the fibers. The action is faithful if the monodromy of the bundle is faithful. In these cases, the isomorphism class of the 4-manifold I believe is determined by the subgroup of homeomorphisms of the circle, up to conjugacy.

For $h > 1$, there is always some branched cover of the base surface so that when you pull the bundle back to the branched cover, there is a section of the bundle, the map to the outer automorphism group of the fiber lifts to the automorphism group, and the fundamental group of the 4-manifold is a semi-direct product.

I'm not an expert in these, and I'm sure there is more that is known.

Answer 2

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.