Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds
The main result of the present paper is the construction of an infinite family of simply connected smooth 4-manifolds, each of which admits at least \(2\) non-isomorphic Lefschetz fibration structures over \({\mathbf S}^{2}\) with the same generic fiber. The only previously known example of such a manifold is due to [\textit{I. Smith}, Symplectic Geometry of Lefschetz fibrations. Ph.D. thesis, University of Oxford (1998)]. The idea of the construction is the following: Start with a simply connected elliptic surface \(E(n)\) and with a well chosen fibered knot \(K\) in \({\mathbf S}^{3}\) of genus \(g\). Performing a knot surgery along an elliptic fiber \(T\) -- using the technique introduced by \textit{R. Fintushel} and \textit{R. Stern} [Invent. Math. 134, No.~2, 363--400 (1998; Zbl 0914.57015)] -- one gets a (simply-connected) symplectic \(4\)-manifold \(E(n)|_{K}\) which admits a genus \(2g+n-1\) Lefschetz fibration structure, as shown by \textit{R. Fintushel} and \textit{R. Stern} [Topology 43, No.~6, 1449--1467 (2004; Zbl 1064.57036)]. By investigating the above Lefschetz fibration structures, the authors show that there are two different momodromy maps of the knot \(K\) which yield two non-isomorphic monodromy factorisations of \(E(n)|_{K}\). This implies the existence of the desired examples for each integer \(n\).
scientific article; zbMATH DE number 5607248
scientific article