Symplectic structure(s) on the Kodaira-Thurston manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:23:06Z http://mathoverflow.net/feeds/question/119630 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119630/symplectic-structures-on-the-kodaira-thurston-manifold Symplectic structure(s) on the Kodaira-Thurston manifold zebaboon 2013-01-23T08:23:46Z 2013-01-23T12:31:00Z <p>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 ($x_{1}, x_{2}, x_{3},x_{4}) \rightarrow (x_{1}+x_{2}, x_{2}, x_{3},x_{4}+1)$</p> <p>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 $T^{2}$.</p> <p>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 $V_{KT}$.</p> <p>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 $(x_{1},x_{4})$ )</p> <p>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 $p'$ (edit: I am almost sure it's not a manifold )</p> <p>My questions are : is the bundle structure I computed with $p$ correct ? Does $p'$ give another bundle structure ?</p> <p>Thank you for your help</p>