Contact structures exist on Brieskorn spheres with non-vanishing hat Heegaard Floer contact invariant but vanishing plus version, proving the two invariants are not equivalent.
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Negative-definite Seifert fibered spaces have a unique negative maximal twisting number, with their fillable tight contact structures induced by Stein structures on the minimal resolution of the underlying complex surface singularity.
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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The hat and plus version of the Heegaard Floer contact invariant are not equivalent
Contact structures exist on Brieskorn spheres with non-vanishing hat Heegaard Floer contact invariant but vanishing plus version, proving the two invariants are not equivalent.
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Fillable structures on negative-definite Seifert fibred spaces
Negative-definite Seifert fibered spaces have a unique negative maximal twisting number, with their fillable tight contact structures induced by Stein structures on the minimal resolution of the underlying complex surface singularity.
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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.