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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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