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Geometric criteria for overtwistedness. (English) Zbl 1408.57026

A contact structure on a \((2n+1)\)-dimensional manifold is a codimension 1 tangent distribution which can be defined by a 1-form \(\alpha\) with \(\alpha\wedge(d\alpha)^n\) nowhere 0. On \(\mathbb R^3\), there are two special contact structures: the standard structure \(\xi_{\text{std}}\) and the standard overtwisted structure \(\xi^1_{\text{std}}\), which are defined, respectively, in cylindrical coordinates \((f,\phi,z)\) by the equations \(dz+\rho^2d\phi=0\) and \(\cos\rho\,dz+\rho\sin\rho\,d\varphi=0\). The disc \(\Delta:=\{z=0,\rho\leq\pi\}\) with the germ of the contact structure \(\xi^1_{\text{std}}\) on it is called the standard overtwisted disc. A contact structure \(\xi\) on a connected 3-manifold is called overtwisted if there is a contact embedding of the standard overtwisted disc \((\Delta,\xi^1_{\text{std}})\) into \((Y,\xi)\). The generalization of an overtwisted disk to a contact structure \(\xi\) on a \((2n+1)\)-dimensional manifold \(Y\) is called a plastikstufe, loosely speaking, a family of overtwisted disks. More rigorously, a plastikstufe with singular set \(S\) is an embedding of \(S\times D^2\) into \(Y\) such that \(S\) is an \({n}\)-dimensional manifold, \(\xi\) induces a singular foliation on \(S\times D^2\), \(\partial (S\times D^2)\) is the only closed leaf of the singular foliation, \(S\times\{0\}\) is the singular set of the foliation and the other leaves of the foliation are diffeomorphic to \(S\times (0,1)\). Note that in 3 dimensions \(S\) must be a point and \(S\times D^2\) is just a disk with singular foliation having a single elliptic point in the center, \(\partial D^2\) is tangent to \(\xi\) and the other leaves spiral from the elliptic point to the boundary. Such a disk is called the standard overtwisted disk. In general one can take the product structure on \(S\times D^2\) so that any point in \(S\) crossed with \(D^2\) has the same foliation as the standard overtwisted disk. Thus a plastikstufe is viewed as a family of overtwisted disks parameterized by \(S\). If \(\Lambda_S\subset(\mathbb R^3,\xi_{\text{std}})\) is the 1-dimensional stabilized Legendrian arc, \(Q\) is a closed manifold, and \(U(Z)\subset T^*Q\) is an open neighborhood of the zero section of the cotangent bundle \((T^*Q,\ker\lambda_{\text{std}})\), then the product smooth submanifold \(\Lambda_S\times Z\subset(B^3\times U(Z),\ker(\alpha_{\text{std}}+\lambda_{\text{std}}))\) is a Legendrian submanifold of the contact structure \(\ker(\alpha_{\text{std}}+\lambda_{\text{std}})\). The contact pair \((B^3\times U(Z),\Lambda_S\times Z)\) endowed with the contact structure \(\ker(\alpha_{\text{std}}+\lambda_{\text{std}})\) is said to be a loose chart. A Legendrian submanifold \(\Lambda\) in a contact manifold \((Y,\xi)\) with \(\dim(Y)\geq 5\) is said to be loose if there is an open set \(V\subset Y\) such that the contact pair \((V,V\cap\Lambda)\) is contactomorphic to a loose chart. For a closed, oriented contact manifold \((Y,\xi)\) and the pair \((S,\varphi)\), let \(K\) be a link in \(Y\). An open book decomposition, denoted by \((S,\varphi)\), for \(Y\) with binding \(K\) is a homeomorphism between \(((S\times[0,1])/\sim_\varphi,(\partial\,S\times[0,1])/\sim_\varphi)\) and \((Y,K)\), where the equivalence relation \(\sim_\varphi\) is defined by \((x,1)\sim_\varphi(\varphi(x),0)\) and \((y,t)\sim_\varphi(y,t')\) for all \(x\in S\), \(y\in\partial\,S\), and \(t,t'\in[0,1]\). The manifold \(Y\) can be identified with \((S\times[0,1])/\sim_\varphi\), and with this identification, \(S_t=S\times\{t\}\), \(t\in[0,1]\) is called a page of the open book decomposition \((S,\varphi)\) and \(\varphi\) is called the monodromy map. An appropriate open book decomposition \((S,\varphi)\) of the smooth manifold \(Y\) determines a contact structure \(\xi\), and conversely every contact manifold \((Y,\xi)\) admits such an adapted open book decomposition. The open books compatible with a contact structure \((Y,\xi)\) induce two contact structures \(\xi_+\) and \(\xi_-\) on \(Y\). The positive stabilization \((Y,\xi_+)\) yields a contact structure contactomorphic to \((Y,\xi)\), but the contact structure \((Y,\xi_-)\) resulting from a negative stabilization is oftentimes not contactomorphic to \((Y,\xi)\).
In this paper, the authors establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, they first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then they characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, they provide several applications of these geometric criteria. The authors show that if \((Y,\xi)\) is a contact manifold of dimension \(2n-1\geq 5\) and \(\alpha_{\text{ot}}\in\Omega^1(\mathbb R^3)\) is a 1-form such that \((\mathbb R^3,\ker\alpha_{\text{ot}})\) is an overtwisted contact structure, then the following conditions are equivalent: (i) the contact structure \((Y,\xi)\) is overtwisted, (ii) there is a contact embedding of \((\mathbb R^3\times\mathbb C^{n-2},\ker(\alpha_{\text{ot}}+\lambda_{\text{st}}))\) into \((Y,\xi)\), (iii) the standard Legendrian unknot \(\Lambda_0\subset(Y,\xi)\) is a loose Legendrian submanifold, (iv) \((Y,\xi)\) contains a small plastikstufe with spherical core and trivial rotation, (v) there exists a contact manifold \((Y',\xi')\) and a loose Legendrian submanifold \(\Lambda\subset(Y',\xi')\) such that \((Y,\xi)\) is contactomorphic to the contact \((+1)\)-surgery of \((Y',\xi')\) along \(\Lambda\), (vi) there exists a negatively stabilized contact open book compatible with \((Y,\xi)\).

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D10 Contact manifolds (general theory)
53D15 Almost contact and almost symplectic manifolds
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