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

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $\mathrm{\Lambda }\subset ({\mathbb{S}}^{2n+1},{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ with $$\mathrm{t}\mathrm{b}(\mathrm{\Lambda })\mathrm{\ne }(-1{)}^{\frac{n(n-1)}{2}+1}$$ 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 $({\mathbb{S}}^3,{\xi }_{\mathrm{s}\mathrm{t}\mathrm{d}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1\mathrm{/}k)$-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.