A landscape of contact manifolds via rational SFT

Abstract: We define a hierarchy functor from the exact symplectic cobordism category to a totally ordered set from a $BL_\infty$ (Bi-Lie) formalism of the rational symplectic field theory (RSFT). The hierarchy functor consists of three levels of structures, namely algebraic planar torsion, order of semi-dilation and planarity, all taking values in $\mathbb{N}\cup {\infty}$, where algebraic planar torsion can be understood as the analogue of Latschev-Wendl's algebraic torsion in the context of RSFT. The hierarchy functor is well-defined through a partial construction of RSFT and is within the scope of established virtual techniques. We develop computation tools for those functors and prove all three of them are surjective. In particular, the planarity functor is surjective in all dimension $\ge 3$. Then we use the hierarchy functor to study the existence of exact cobordisms. We discuss examples including iterated planar open books, spinal open books, affine varieties with uniruled compactification and links of singularities.