| In this thesis we study four and six dimensional nilmanifolds using their associated rational Lie algebras and minimal models. We show that all four dimensional nilmanifolds have symplectic structures. We then show that there exists a family of four dimensional nilmanifolds, non diffeomorphic to the Kodaira-Thurston manifold, which fibrate symplectically as torus bundles over tori. Using similar methods we also investigate which six dimensional nilmanifolds possess symplectic structures. Our last result concerns symplectic torus actions. We show that the Duistermaat-Heckman function defined on a torus is a piecewise trignometric polynomial. We present examples of torus valued moment maps on a family of symplectic manifolds studied by Cordero, Fernandez and Gray. |
