Topologically distinct Lagrangian and symplectic fillings
In some cases, the smooth topology of a symplectic manifold is defined by the contact topology of its boundary. For example, in [Lond. Math. Soc. Lect. Note Ser. 151, 45--72 (1990; Zbl 0731.53036)], \textit{Y.~Eliashberg} proved that any symplectically aspherical filling of the standard contact \(S^3\) is diffeomorphic to \(B^4\). \textit{B. Chantraine} in [Algebr. Geom. Topol. 10, No. 1, 63--85 (2010; Zbl 1203.57010)], showed that if \(L\) is an orientable Lagrangian filling of an oriented Legendrian link \(\Lambda\) in the boundary of a Stein surface, then \(\text{tb}(\Lambda)=-\chi(L)\), and if \(\Lambda\) is a knot, then the topology of a Lagrangian filling of \(\Lambda\) is completely determined by its Thurston-Bennequin invariant \(\text{tb}(\Lambda)\). Based on this result, in this paper the authors are interested if the topology of a Lagrangian filling of an oriented Legendrian link in \(S^3\) is completely determined by its Thurston-Bennequin invariant. The authors show that, in contrast to the the case of Legendrian knots, the topology is not defined by the tb-invariant, and they show that for every integer \(N>1\), there exists a Legendrian link \(\Lambda\subset(S^3,\xi_0)\) with \(p(N)\) non-homeomorphic exact orientable Lagrangian fillings, where \(p(N)\) denotes the number of partitions of \(N\). Also, they show that for every integer \(N>1\), there exists a link \(K\subset S^3\) with \(p(N)\) non-singular complex algebraic curves that intersect \(S^3\) transversally in \(K\) with the property that the intersections of the curves with \(B^4\) are pairwise non-homeomorphic. Finally, the authors extend the first result to higher dimensions and they show that for all integers \(n\), \(N>1\), there exists a connected Legendrian submanifold \(\Lambda\subset(S^{2n+1},\xi_0)\) with \(p(N)\) non-homeomorphic connected exact orientable Lagrangian fillings.
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