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**Topological complexity of symplectic 4-manifolds and Stein fillings.**
*(English)*
Zbl 1348.57035

If \(\Sigma^s_{g,r}\) is a compact oriented surface of genus \(g\) with \(s\) boundary components and \(r\) marked points in its interior, then the mapping class group \(\Gamma^s_{g,r}\) of \(\Sigma^s_{g,r}\) is the group of isotopy classes of orientation-preserving self-diffeomorphisms of \(\Sigma^s_{g,r}\), fixing the points on the boundary and fixing each marked point individually. A positive Dehn twist along an embedded simple closed curve \(\alpha\) on \(\Sigma^s_{g,r}\) is \(t_\alpha\) and a negative Dehn twist is \(t_\alpha^{-1}\). The group \(\Gamma^s_{g,r}\) is generated by Dehn twists. An element \(\Phi\in\Gamma^s_{g,r}\) that can be expressed as a product of positive Dehn twists is called a positive factorization of \(\Phi\). A Lefschetz fibration is a surjective map \(f:X\to\Sigma\), where \(X\) and \(\Sigma\) are \(4\)- and \(2\)-dimensional compact manifolds, respectively, such that \(f\) fails to be a submersion along a non-empty discrete set \(C\), and around each critical point in \(C\) it conforms to the local model \(f(z_1,z_2)=z_1\,z_2\), compatible with orientations. All the Lefschetz fibrations will be over \(\Sigma=\mathbb S^2\). A Lefschetz pencil is defined as a Lefschetz fibration on the complement of a discrete set \(B\) in \(X\), where \(f\), around each base point in \(B\), conforms to the local model \(f(z_1,z_2)=z_1/z_2\). An open book decomposition \({\mathcal B}\) of a \(3\)-manifold \(Y\) is a pair \((L,f)\), where \(L\) is an oriented link in \(Y\), called the binding, and \(f:Y\setminus L\to S^1\) is a fibration such that \(f^{-1}(t)\) is the interior of a compact oriented surface \(F_t\subset Y\) and \(\partial F_t=L\) for all \(t\in S^1\). A contact structure \(\xi\) on a \(3\)-manifold \(Y\) is said to be supported by an open book \({\mathcal B}=(L,f)\) if \(\xi\) is isotopic to a contact structure given by a \(1\)-form \(\alpha\) satisfying \(\alpha >0\) on positively oriented tangents to \(L\) and \(d\alpha\) is a positive volume form on every page. It is said that the open book \((L,f)\) is compatible with the contact structure \(\xi\) on \(Y\). A compact 4-manifold \(X\) with non-empty connected boundary \(\partial X\) is said to be a Stein filling of a closed contact \(3\)-manifold \((M,\xi)\) if \(X\) is the sub-level set of a plurisubharmonic function on a Stein surface and the contact structure on \(\partial X\) induced by the complex tangencies is contactomorphic to \((M, \xi)\).

In this paper, the authors show that there exists no a priori bound on the Euler characteristic of a closed symplectic \(4\)-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact \(3\)-manifold coming from the genus of a compatible open book. The examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary are presented. For \(g\geq 11\), the authors prove specifically that (i)the positive Dehn multitwist along the boundary can be factorized as a product of arbitrarily large number of positive Dehn twists along non-separating curves in \(\Gamma^2_{g,r}\), (ii)there is a family of relatively minimal genus \(g\) Lefschetz pencils \(\{(X_m,f_m);\;m\in\mathbb N\}\) such that the Euler characteristic of the closed symplectic \(4\)-manifold \(X_m\) is strictly increasing in \(m\), and (iii) there are infinitely many closed \(3\)-manifolds admitting genus \(g\) open books with connected binding which bound allowable Lefschetz fibrations over the \(2\)-disk with arbitrarily large Euler characteristics.

In this paper, the authors show that there exists no a priori bound on the Euler characteristic of a closed symplectic \(4\)-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact \(3\)-manifold coming from the genus of a compatible open book. The examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary are presented. For \(g\geq 11\), the authors prove specifically that (i)the positive Dehn multitwist along the boundary can be factorized as a product of arbitrarily large number of positive Dehn twists along non-separating curves in \(\Gamma^2_{g,r}\), (ii)there is a family of relatively minimal genus \(g\) Lefschetz pencils \(\{(X_m,f_m);\;m\in\mathbb N\}\) such that the Euler characteristic of the closed symplectic \(4\)-manifold \(X_m\) is strictly increasing in \(m\), and (iii) there are infinitely many closed \(3\)-manifolds admitting genus \(g\) open books with connected binding which bound allowable Lefschetz fibrations over the \(2\)-disk with arbitrarily large Euler characteristics.

Reviewer: Andrew Bucki (Edmond)

### MSC:

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

53D35 | Global theory of symplectic and contact manifolds |