  Let $F',F$ be any two closed orientable surfaces of genus $g'>g\ge 1$, and
$f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a
pseudo-Anosov map $f':F'\to F'$ so that there is a fiber preserving degree one
map $M(F',f')\to M(F,f)$ between the hyperbolic surface bundles. Moreover the
extension $f'$ can be chosen so that the surface bundles $M(F',f')$ and
$M(F,f)$ have the same first Betti numbers.
