  Let $g$ be a non-negative integer, $\Sigma _g$ a closed orientable surface of
genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the
group homomorphisms $\pi _1(\Sigma _g)\to G$ up to the action of
$\mathcal{M}_g$ on $\pi _1(\Sigma _g)$ in the following cases; (1)
$G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case
(2), we completely classify orientable $T^2$-bundles over closed orientable
surfaces up to bundle isomorphisms. In particular, we show that any orientable
$T^2$-bundle over $\Sigma _g$ with $g\geq 1$ is isomorphic to the fiber
connected sum of $g$ pieces of $T^2$-bundles over $T^2$. Moreover, the
classification result in the case (1) can be generalized into the case where
$G$ is the free product of finite number of finite cyclic groups. We also apply
it to an extension problem of maps from a closed surface to the connected sum
of lens spaces.
