On contact 3-manifolds that admit a nonfree toric action

Abstract

We classify all contact structures on 3-manifolds that admit a nonfree toric action, up to contactomorphism, and present them through explicit topological descriptions. Our classification is based on Lerman's classification of toric contact 3-manifolds up to equivariant contactomorphism [Lerman, J. Symplectic Geom. 1 (2003), 785–828]. We also prove that every contact 3-manifold with a nonfree toric action arises as the concave boundary of a toric linear plumbing over spheres inspired by Marinković et al. As a corollary of both results, we classify which contact 3-manifolds arise as the concave boundary of a linear plumbing of spheres.