Loose Legendrians and the plastikstufe
The notion of overtwisted contact structures on three-dimensional manifolds introduced by \textit{Y. Eliashberg} [Invent. Math. 98, No. 3, 623--637 (1989; Zbl 0684.57012)] played an important role in the study of contact topology. A contact structure \(\xi\) on a \(3\)-dimensional manifold \(M\) is called \textit{overtwisted} if it contains an \textit{overtwisted disc}, which is an embedded closed disc \(D\subset M\) whose boundary \(\partial D\) is not only Legendrian for \(\xi\) but also transverse to \(\xi\) along \(\partial D\). Such contact structures on closed \(3\)-manifolds exhibit very nice flexibility: their classification up to isotopy coincides with their homotopy classification as plane fields. In other words they satisfy an \(h\)-principle. \textit{K. Dymara} [Ann. Global Anal. Geom. 19, No. 3, 293--305 (2001; Zbl 0985.57009)] showed that any Legendrian knot in the complement of an overtwisted disk also satisfies an \(h\)-principle.NEWLINENEWLINEThe paper under review gives a higher-dimensional generalization for the last claim. Recall that as a generalization of the overtwisted disk to higher dimensions \textit{K. Niederkrüger} [Algebr. Geom. Topol. 6, 2473--2508 (2006; Zbl 1129.53056)] introduced the notion `plastikstufe'. Let \(B\) be a closed \((n-1)\)-manifold. A \textit{plastikstufe \({\mathcal P}_B\) with core \(B\)} in a \((2n+1)\)-dimensional contact manifold \((M, \xi)\) is a submanifold \({\mathcal P}_B=D^2\times B\hookrightarrow M\) such that \(\xi\cap T{\mathcal P}_B\) is a singular foliation that is tangent to the fibers \(\{z\}\times B\) for every \(z\in D^2\), and that restricts on every slice \(D^2\times\{b\}\) to the foliation of the overtwisted disk sketched. A plastikstufe \({\mathcal P}_B\) is called a \textit{small plastikstufe} if there is an embedded open ball in \((M, \xi)\) containing \({\mathcal P}_B\). A contact manifold that contains a plastikstufe is called \textit{PS-overtwisted}. A \textit{leaf ribbon} in a plastikstufe \({\mathcal P}_B\) with core \(B\) is a thin ribbon diffeomorphic to \(B\times (0,1)\), obtained by shrinking a leaf of \({\mathcal P}_B\) within itself into the standard neighborhood of the core \(B\). A small plastikstufe as above is said to have \textit{spherical core} (resp. \textit{trivial rotation}) if \(B=S^{n-1}\) (resp. the relative rotation class between a leaf ribbon of \({\mathcal P}_B\) and a punctured Legendrian disk vanishes). \textit{Y. Eliashberg} and \textit{M. Fraser} [J. Symplectic Geom. 7, No. 2, 77--127 (2009; Zbl 1179.57040)] called a Legendrian knot in a \(3\)-dimensional contact manifold \textit{loose} if its complement is overtwisted. Similarly, \textit{E. Murphy} [``Loose Legendrian Embeddings in High Dimensional Contact Manifolds'', \url{arXiv:1201.2245}] defined a \textit{Legendrian knot} in a contact manifold \((M^{2n+1},\xi)\) as a closed, connected, embedded submanifold \(L^n\hookrightarrow M^{2n+1}\) such that \(TL\subset\xi\), and called such a Legendrian knot (with \(n>1\)) \textit{loose} if it contains a sufficiently thick Weinstein neighborhood of a stabilized Legendrian curve. The main result of the paper under review claims: if a contact manifold \((M^{2n+1}, \xi)\) contains a small plastikstufe with spherical core and trivial rotations then any Legendrian submanifold in the complement of this plastikstufe is loose, and hence flexible. (Actually, a local version of the above result by K. Dymara in the three-dimensional situation was also given.) Combining this result with the symplectic \(h\)-cobordism theorems in [\textit{K. Cieliebak} and \textit{Y. Eliashberg}, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. Colloquium Publications. American Mathematical Society 59. Providence, RI: American Mathematical Society (AMS). (2012; Zbl 1262.32026)] the authors obtain: For two contact structures \(\xi_0,\xi_1\) on a manifold \(Y\) of dimension \(2n+1>3\), and a simply connected contact manifold \((M^{2n+1}, \xi_{\mathrm PS})\) containing a small plastikstufe with spherical core and trivial rotation, \((M,\xi_{\mathrm PS})\sharp(Y,\xi_0)\) is contactomorphic to \((M,\xi_{\mathrm PS})\sharp(Y,\xi_1)\) provided that one of the following conditions holds:NEWLINENEWLINE \(\bullet\) There exists a manifold \(W\) of dimension \(2n+2\) with \(\partial W=Y\), and \(W\) carries two Stein structures \(J_0, J_1\) such that \((W, J_0)\) is a filling for \(\xi_0\), \((W, J_1)\) is a filling for \(\xi_1\), and \(J_0, J_1\) are homotopic through almost complex structures.NEWLINENEWLINE \(\bullet\) There exists a Stein cobordism \((W,J)\) from \((Y,\xi_0)\) to \((Y,\xi_1)\) such that \(W\) is smoothly the product cobordism \(Y\times [0,1]\).
scientific article; zbMATH DE number 6185366
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