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Corollary 1.8 of this paper implies that the Pontryagin number for a four-dimensional surface bundle is non-zero only when the surface has genus $g > 2$. I would like to ask the following question:

What is the minimal value of this non-zero Pontryagin number and for which surface bundle?

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As the first Pontryagin number of a closed oriented four-manifold is three times the signature, your question can be rephrased as follows:

What is the minimal value of the signature of a four-dimensional surface bundle and which surface bundles realise it?

Theorem A of this paper states that if $F \to E \to B$ is a fiber bundle of closed connected PL manifolds which are compatibly oriented, then $\sigma(E) \equiv \sigma(F)\sigma(B) \bmod 4$. Therefore, the signature of an oriented surface bundle over a surface is divisible by four. Moreover, in Die Signatur von Flächenbündeln, Meyer showed that for every $h \geq 3$ and $n \in \mathbb{Z}$, there is $g \geq 0$ and a surface bundle $\Sigma_h \to X \to \Sigma_g$ with $\sigma(X) = 4n$.

So the signature of a surface bundle over a surface can be arbitrarily negative; in particular, there is no minimum value. If you're asking for the smallest possible value of $|\sigma(X)|$ which is non-zero, then the answer is $4$. An example of such a surface bundle is given by Endo in this paper where he constructs a bundle $\Sigma_3 \to X \to \Sigma_{111}$ with $\sigma(X) = -4$.

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