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Symplectic fillability of tight contact structures on torus bundles. (English) Zbl 0974.53061
A manifold $$M$$ with contact structure $$\xi$$ is weakly symplectically fillable if $$M = \partial W$$, where $$(W, \omega)$$ is a symplectic manifold such that $$\omega|_\xi$$ is nondegenerate (and orientations are compatible). $$M$$ is stongly symplectically fillable if $$M = \partial W$$, $$(W, \omega)$$ symplectic, and there is a Liouville vector field $$X$$ (outward on $$\partial W$$) with $$\xi = \text{ker}(i_X\omega|_M)$$. Work of Giroux and Eliashberg describes when tight contact structures on $$T^3$$ are weakly or strongly symplectically fillable. In this paper, the same type of description is given for $$T^2$$-bundles over $$S^1$$ constructed from elements of $$\text{SL}_2(\mathbb Z)$$ with certain associated tight contact structures $$\xi_n$$.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 57M50 General geometric structures on low-dimensional manifolds 57R65 Surgery and handlebodies
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