In paper arXiv:math/0701247
*"Divisibility of the stable Miller-Morita-Mumford classes"* by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown
that the Pontryagin numbers for a 4-dim surface bundles are
$0$ mod $D$, where $D=12$ is the maximal divisor of the Pontryagin numbers.


**If we fix the fiber of the 4-dim surface bundle to be a genus $g$
surface, then what is the maximal divisor of the Pontryagin numbers for each $g$?**
I think we know that Pontryagin number is always zero for $g=0, 1$.



Pontryagin number for 4-dim surface bundle
Ulrike Tillmann told me that Endo ( <http://projecteuclid.org/euclid.ojm/1200788349> 1998) proved that:
if the genus of the fibre $g>2$, then
there is a surface bundle over a surface of a genus equal or less than 111, and the surface bundle has a signature $\pm 4$, hence realising the minimal Ponrtryagin number 12 for each $g>2$. 


