Contact 3-manifolds that admit a non-free toric action

TL;DR

This work completes the classification of contact toric 3-manifolds admitting a non-free toric action up to contactomorphism. It separates the tight and overtwisted cases, establishing that each lens space $L(k,l)$ has a unique universally tight toric structure (as a quotient of the tight $S^3$ structure) and exactly two toric overtwisted structures, obtainable by half- and full-Lutz twists. A central contribution is proving a converse to a prior result: every such non-free toric contact 3-manifold can be realized as the concave boundary of a linear plumbing of sphere bundles with nonnegative self-intersections, with an explicit continued-fraction procedure to construct the plumbing. The results bridge contact toric geometry with 4-dimensional plumbings, providing explicit geometric realizations and contributing to the understanding of when toric structures arise from topological plumbing constructions. This has potential implications for extending these classification methods to related toric and contact-geometric settings in higher dimensions.

Abstract

We classify contact toric 3-manifolds up to contactomorphism, through explicit descriptions, building off of work by Lerman [Lerman03]. As an application, we classify all contact structures on 3-manifolds that can be realised as a concave boundary of linear plumbing over spheres. The later result is inspired by the work [MNRSTW25].

Figures (5)

• Figure 1: A moment cone of $(Y(t_1,t_2),\xi_{t_1,t_2})$
• Figure 2: L-shape- the moment map image of $(s_1,s_2)$
• Figure 3: Gluing of $(0,s_n)$ to $(0,s_{n-1})$
• Figure 4: Moment cone of a Lens space $L(k,l)$ with a standard tight contact structure, left $l>0$, right $k=0.$
• Figure 5: a) A half-Lutz twist, b) a full-Lutz twist

Theorems & Definitions (20)

• Corollary 1.5
• Example 2.1
• Definition 2.2
• Remark 2.4
• proof : Proof of Theorem \ref{['thm:tight']}
• Definition 4.1
• Proposition 4.2
• proof
• proof : Proof of Theorem \ref{['thm:overtwisted']}.
• Proposition 4.3