We generalize the familiar notions of overtwistedness and Giroux torsion in -dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order can be interpreted as measuring a gradation in “degrees of tightness” of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact-type embeddings into any closed symplectic -manifold, and has vanishing contact invariant in embedded contact homology, and we give examples of contact manifolds that have planar -torsion for any but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness, and compactness theorems for certain classes of -holomorphic curves in blown-up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains.
"A hierarchy of local symplectic filling obstructions for contact -manifolds." Duke Math. J. 162 (12) 2197 - 2283, 15 September 2013. https://doi.org/10.1215/00127094-2348333