Some applications of Menke's JSJ decomposition for symplectic fillings

Austin Christian, Youlin Li

Published 2020-06-30Version 1

We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a symplectic 2-handle to a filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying symplectic fillings to the same problem for universally tight contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.