Algebraic overtwistedness of contact $(+1)$-surgeries via augmented Floer theories

Abstract

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $Λ\subset(\mathbb{S}^{2n+1},ξ_{\mathrm{std}})$ with \[ \mathrm{tb}(Λ)\ne (-1)^{\frac{n(n-1)}{2}+1} \] has vanishing contact homology. In dimension three, this produces new examples of tight algebraically overtwisted contact manifolds, vastly extending the authors' previous results: If a Legendrian knot in $(\mathbb{S}^3,ξ_{\mathrm{std}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1/k)$-surgery on it is algebraically overtwisted and tight for every positive integer $k$. Our proofs use versions of symplectic cohomology and wrapped Floer cohomology defined by augmentations of a chain-level contact homology algebra.