Authors derive a closed-form formula for Gompf's θ-invariant of canonical contact structures on Seifert fibered 3-manifolds and a recursive formula for general plumbing trees using Legendrian surgery descriptions.
Fillable structures on negative-definite Seifert fibred spaces
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abstract
We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, we show that such Seifert fibred spaces admit a unique negative maximal twisting number, and compute it explicitly using the Alexander filtration in lattice cohomology. In particular, we show that the negative-twisting tight structures on these manifolds are induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. As an application, we provide a necessary condition for a negative-definite Seifert fibred space to admit a separating contact-type embedding in a strong symplectic filling of a generalised $L$-space.
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math.GT 4years
2026 4roles
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Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures
Mazur manifolds with boundaries Σ(2,3,13), Σ(2,5,7), and Σ(3,4,5) admit no symplectic structure, producing exotic pairs of simply connected 4-manifolds with definite intersection forms.
Provides the complete list of Brieskorn spheres carrying at most two symplectically fillable contact structures up to isotopy.
citing papers explorer
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The Gompf $\theta$-Invariant of Canonical Contact Structures via Legendrian Surgery
Authors derive a closed-form formula for Gompf's θ-invariant of canonical contact structures on Seifert fibered 3-manifolds and a recursive formula for general plumbing trees using Legendrian surgery descriptions.
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Brieskorn spheres and rational homology ball symplectic fillings
Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures
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Mazur manifolds and symplectic structures
Mazur manifolds with boundaries Σ(2,3,13), Σ(2,5,7), and Σ(3,4,5) admit no symplectic structure, producing exotic pairs of simply connected 4-manifolds with definite intersection forms.
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Brieskorn spheres with two fillable contact structures
Provides the complete list of Brieskorn spheres carrying at most two symplectically fillable contact structures up to isotopy.