# Does there exist a surface bundle over a surface of genus at least 2 that fibers in three distinct ways?

Let $$\Sigma_g \to E \to \Sigma_h$$ be a surface bundle over a surface. Unless otherwise stated, I'll assume $g, h \ge 2$. The theory of Thurston norm shows that surface bundles over $S^1$ often fiber in infinitely many ways (e.g. with fibers of infinitely many genera). For surface bundles over surfaces, however, the Euler characteristic (which is the product of the characteristics of the base and the fiber) provides an arithmetic constraint on the possible genera of the base and the fiber. When the genus of the base and the fiber are both at least two, this shows that there are only finitely many possible fiber genera, and an analysis of the fundamental group shows that there are only finitely many possibilities for the subgroup of $\pi_1E$ corresponding to the fiber (this follows, for instance, by work of F.E.A. Johnson). Moreover, Johnson's work also shows that when the monodromy representation $\rho: \pi_1 \Sigma_h \to \operatorname{Mod}(\Sigma_g)$ is non-injective, there are at most two fiberings, so that any bundle fibering in at least three ways must have injective monodromy.

I am aware of the example of Atiyah-Kodaira, which shows that it is possible for bundles with injective monodromy to fiber in at least two ways (of course, product bundles are a trivial example of multiply-fibered total spaces as well). However, I haven't seen any example of a bundle with three distinct fiberings when the base genus is at least two. When the base genus is one, there are trivial constructions one can do: if $M$ is any three-manifold fibering over $S^1$ in infinitely many ways, then $M\times S^1$ will fiber over $S^1\times S^1$ in infinitely many ways. I don't know of an example of a torus bundle over a surface that fibers in infinitely many ways, but I would be interested to see this, too.

With all this said, does anyone know of an example of a surface bundle over a surface of genus $h \ge 2$ that fibers in at least three ways (i.e these are pairwise non-fiberwise diffeomorphic)? What about a surface bundle over a closed $2k$-manifold of nonzero Euler characteristic that fibers in at least $k+2$ ways? (Note that a product of $k+1$ surfaces gives an example where there are $k+1$ fiberings).

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Initially, I thought that the $M\times S^1$ example of a bundle over a torus fibering in infinitely many ways might be used as a starting point in a construction of a bundle with multiple fiberings over a surface of higher genus, say by pulling back along a branched cover of $T^2$. But this turns out not to work: the monodromy will then factor as $\pi_1 \Sigma_g \to \mathbb Z ^2 \to \operatorname{Mod}(\Sigma_g)$, and as the first map certainly cannot be injective, Johnson's result then prevents this bundle from having more than two fiberings. –  Nick Salter Feb 15 '13 at 22:55
See Theorem 2.10 of this paper for some discussion: front.math.ucdavis.edu/1106.4595 –  Ian Agol Feb 27 '13 at 20:09

Yes. I recently found methods of constructing surface bundles over surfaces with at least $n$ fiberings for any $n$. The idea is to perform a fiberwise connect-sum of trivial bundles in such a way that the two different fiberings for each component can be compatibly assembled into a bundle with many fiberings. See this preprint.