Classification of orientable torus bundles over closed orientable surfaces

Let g be a non-negative integer, g a closed orientable surface of genus g, and Mg its mapping class group. We classify all the group homomorphisms ₁ (g) G up to the action of Mg on ₁ (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 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.