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I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open.

Is there a (smooth) surface bundle over a surface $\Sigma_h \to E \to \Sigma_g$ that does not admit a flat structure? (Equivalently, one can ask whether there is a homomorphism $\pi_1(\Sigma_g) \to \pi_0 \text{Diff}(\Sigma_h)$ that cannot be lifted to the group of diffeomorphisms.)

What is the status of this question, and could somebody point out a reference that mentions this as an open question? Finally, is something known for small values of $g,h$ (but still $g,h \geq 2$)?

(In case you do not understand what I am asking, I can very much recommend reading the last chapter of Morita's book Geometry of Characteristic Classes.)

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It's still open for all values of $g$ and $h$. One reference for it is

M. Bestvina, T. Church, and J. Souto, Some groups of mapping classes not realized by diffeomorphisms, Comment. Math. Helv. 88 (2013), no. 1, 205–220.

It is stated as an open problem at the end of the introduction.

One interesting positive result in this direction is in

D. Kotschick and S. Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005), no. 1, 131–149.

which proves that every surface bundles over a surface can be given a flat structure after possibly fiber-connect-summing with a trivial bundle.

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