Classification of orientable torus bundles over closed orientable surfaces

Let g be a non-negative integer, Σ g a closed orientable surface of genus g, and Mg its mapping class group. We classify all the group homomorphisms π ₁(Σ g)→ G up to the action of Mg on π ₁(Σ g) in the following cases; (1) G=PSL(2;Z), (2) G=SL(2;Z). As an application of the case (2), we completely classify orientable T²-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable T²-bundle over Σ g with g≥ 1 is isomorphic to the fiber connected sum of g pieces of T²-bundles over T². 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.