RSFT functors for strong cobordisms and applications

TL;DR

This paper develops new functors for strong symplectic cobordisms, proving properties of algebraic torsion and exploring contact 3-folds, with implications for symplectic and contact topology.

Contribution

It extends hierarchy functors to strong cobordisms using Maurer--Cartan deformations and establishes new results on algebraic torsion and contact manifolds.

Findings

Concave boundary has finite algebraic planar torsion if convex boundary does.

Existence of contact 3-folds without strong cobordisms to the standard sphere.

Generalizations relating algebraic torsion to holomorphic curve invariants.

Abstract

We extend the hierarchy functors of [33] to the case of strong symplectic cobordisms, via deformations with Maurer--Cartan elements. In particular, we prove that the concave boundary of a strong cobordism has finite algebraic planar torsion if the convex boundary does, which yields a functorial proof of finite algebraic planar torsion for contact manifolds admitting strong cobordisms to overtwisted contact manifolds. We also show the existence of contact $3$-folds without strong cobordisms to the standard contact $3$-sphere, that are not cofillable. We also include generalizations of the theory relating our notion of algebraic planar torsion to Latschev--Wendl's notion of algebraic torsion, discussing variations from counting holomorphic curves with general constraints and invariants extracted from higher genera holomorphic curves from an algebraic perspective.