  We generalize the familiar notions of overtwistedness and Giroux torsion in
3-dimensional contact manifolds, defining an infinite hierarchy of local
filling obstructions called planar torsion, whose integer-valued order $k \ge
0$ 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
4-manifold, and has vanishing contact invariant in Embedded Contact Homology,
and we give examples of contact manifolds that have planar k-torsion for any $k
\ge 2$ 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 J-holomorphic curves in blown up summed open books; these
also imply algebraic obstructions to planarity and embeddings of partially
planar domains. The results are applied further in followup papers on weak
symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact
symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory
(arXiv:1009.3262, joint with J. Latschev).
