Let $V_{KT}$ be the Kodaira-Thurston 4-manifold $\frac{\mathbb{R^{4}}}{G}$ where $G$ is the subgroup of $Diff(\mathbb{R^{4}})$ generated by unit translations along the $x_{1}$, $x_{2}$, $x_{3}$ axis and ($x_{1}, x_{2}, x_{3},x_{4}) \rightarrow (x_{1}+x_{2}, x_{2}, x_{3},x_{4}+1)$
The projection $p : V_{KT} \rightarrow T^{2}$ along the $(x_{2},x_{3})$ factor gives $V_{KT}$ a strucure of bundle over $T^{2}$ with fibers diffeomorphic to $T^{2}$.
The symplectic form $dx_{1} \wedge dx_{2}+ dx_{3} \wedge dx_{4}$ on $\mathbb{R^{4}}$ is invariant by the action of $G$ on $\mathbb{R^{4}}$ hence descends to a symplectic form $\omega$ on $V_{KT}$.
Consequently $p$ gives $V_{KT}$ a bundle structure over $T^{2}$ (in coordinates $(x_{2},x_{3})$ ) whose fiber is symplectomorphic to a Lagrangian $T^{2}$ (in coordinates $(x_{1},x_{4})$ )
However often in literature the projection $p' : V_{KT} \rightarrow T^{2}$ along the $(x_{3},x_{4})$ factor is considered and i have trouble to compute the fiber of $p'$ (is it again edit: I am almost sure it's not a torus ? manifold )
My questions are : is the bundle structure I computed with $p$ correct ? Does $p'$ give another bundle structure , and in this case, is the fiber a torus ?
Thank you for your help
