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**Tight contact structures on fibered hyperbolic 3-manifolds.**
*(English)*
Zbl 1083.53082

The paper studies tight contact structures \(C\) on a closed hyperbolic \(M^3\) that fibers over \(S^1\) with the fiber \(\Sigma_g, g\geq 2.\) Let \(e_C\in H^2(M)\) be the Euler class of such tight \(C\) and let \([\Sigma_g]\in H_2(M)\) be the homology class of a fiber. Ya. Eliashberg proved that \(-(2g-2)\leq e_C([\Sigma_g])\leq 2g-2\), see: [Ann. Inst. Fourier 42, No.1–2, 165–192 (1992; Zbl 0756.53017)].

The authors prove that if the monodromy map \(\Sigma\to \Sigma\) is pseudo-Anosov, then there exists a unique (up to isotopy) tight contact structure \(C\) on \(M\) satisfying the extremal case \(e_C([\Sigma_g])=2g-2\). Similarly, there exists a unique tight \(C\) satisfying the other extremal case \(e_C([\Sigma_g])=-(2g-2).\) These contact structures are weakly symplectically fillable and universally tight. Every \(C^0\)-small perturbation of the fibration into a contact structure is isotopic to one of these two contact structures.

This nice theorem follows from the relative classification of tight contact structures on \(N=\Sigma_g\times [0,1], g \geq 2\) obtained in the paper. Fix \(\Gamma_{\Sigma_g^i}=2\gamma_i\subset \Sigma_g^i =\Sigma_g\times i\subset \Sigma_g\times [0,1], i=0,1\) that consists of two parallel copies of a nonseparating curve \(\gamma_i, i=0,1.\) Choose a characteristic foliation \(\mathfrak F\) on \(\partial M\) that is divided by \(\Gamma_{\Sigma_g^0}\sqcup \Gamma_{\Sigma_g^1}\) into positive and negative regions, i.e., regions where the normal orientation of the surface respectively agrees and does not agree with the normal orientation of the contact structure (to be constructed). Assume moreover that the two annular regions have the same sign. (Otherwise the corresponding contact structure does not exist.) Then all contact structures that satisfy the boundary condition \(\mathfrak F\) are universally tight. If \(\gamma_0\neq \gamma_1\) then there are \(4\) such contact structures on \(N\), and if \(\gamma_0=\gamma_1\) then there are \(5\) such contact structures. In many cases these structures are distinguishable by their relative Euler class.

The authors prove that if the monodromy map \(\Sigma\to \Sigma\) is pseudo-Anosov, then there exists a unique (up to isotopy) tight contact structure \(C\) on \(M\) satisfying the extremal case \(e_C([\Sigma_g])=2g-2\). Similarly, there exists a unique tight \(C\) satisfying the other extremal case \(e_C([\Sigma_g])=-(2g-2).\) These contact structures are weakly symplectically fillable and universally tight. Every \(C^0\)-small perturbation of the fibration into a contact structure is isotopic to one of these two contact structures.

This nice theorem follows from the relative classification of tight contact structures on \(N=\Sigma_g\times [0,1], g \geq 2\) obtained in the paper. Fix \(\Gamma_{\Sigma_g^i}=2\gamma_i\subset \Sigma_g^i =\Sigma_g\times i\subset \Sigma_g\times [0,1], i=0,1\) that consists of two parallel copies of a nonseparating curve \(\gamma_i, i=0,1.\) Choose a characteristic foliation \(\mathfrak F\) on \(\partial M\) that is divided by \(\Gamma_{\Sigma_g^0}\sqcup \Gamma_{\Sigma_g^1}\) into positive and negative regions, i.e., regions where the normal orientation of the surface respectively agrees and does not agree with the normal orientation of the contact structure (to be constructed). Assume moreover that the two annular regions have the same sign. (Otherwise the corresponding contact structure does not exist.) Then all contact structures that satisfy the boundary condition \(\mathfrak F\) are universally tight. If \(\gamma_0\neq \gamma_1\) then there are \(4\) such contact structures on \(N\), and if \(\gamma_0=\gamma_1\) then there are \(5\) such contact structures. In many cases these structures are distinguishable by their relative Euler class.

Reviewer: Vladimir Chernov (Hanover)

### MSC:

53D35 | Global theory of symplectic and contact manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |