Generalizations of planar contact manifolds to higher dimensions

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions.We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer $m \gt 3$, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension $m$. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.