Is the following statement correct or known to be correct?


**For a 4-dimensional closed orientable
surface bundle $E$ with fiber of genus 2, the signature must be 0 mod 8 (or the Pontryagin number of $E$ must be 0 mod 24).**


I think it is known that for a 4-dimensional closed orientable
surface bundle $E$, the Pontryagin number of $E$ must be 0 mod 12
(see [Pontryagin number for 4-dim surface bundle](https://mathoverflow.net/questions/165918) )



the Pontryagin number of a 4-dim orientable surface bundle with fiber of genus 2
In fact, the signature must equal $0$.


Let $\pi: E \rightarrow B$ be the bundle in question, so $B$ is a closed orientable surface and the fibers of $\pi$ are genus $2$ surfaces. Letting $M\_g$ be the genus $g$ mapping class group, the bundle $\pi$ is classified by a monodromy homomorphism $f:\pi\_1(B) \rightarrow M\_2$. For all $k \geq 1$, there are elements $e\_k \in H^{2k}(M\_2;\mathbb{Q})$ called the Miller-Mumford-Morita (MMM) classes (they are defined in a uniform way for all mapping class groups). The signature of $E$ is equal to $1/3$ of the value of 
$$f^{\ast}(e\_1) \in H^2(B;\mathbb{Q}) = \mathbb{Q};$$
see Proposition 4.11 in Morita's "Geometry of Characteristic Classes". Up to now, everything would work the same no matter what the genus of the fiber is.


We now come to the main point: Igusa proved that $H^{k}(M\_2;\mathbb{Q}) = 0$ for all $k \geq 1$, so $e\_1 = 0$ and the signature vanishes.


