# The Pontryagin number for 4-dim surface bundle

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?

## 1 Answer

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 $$\sigma(X) = -4$$.