Hyperelliptic Lefschetz fibrations and branched covering spaces (Q5929046)

As it is known, the existence of smooth Lefschetz fibrations provides a purely topological description of symplectic 4-manifolds: see, in particular, the important results by \textit{S. K. Donaldson} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, Vol. II, 309-314 (1998; Zbl 0909.53018)] and \textit{R. E. Gompf} and \textit{A. I. Stipsicz} [4-manifolds and Kirby calculus, Grad. Stud. Math. 20 (1999; Zbl 0933.57020)]. The present paper takes into account smooth 4-manifolds \(M\) which admit a relatively minimal hyperelliptic Lefschetz fibration over \(\mathbb{S}^2\). In this setting, the main results achieved are:NEWLINENEWLINENEWLINE\(\bullet\) if all vanishing cycles for the fibration are nonseparating curves, than \(M\) is a 2-fold cover of an \(\mathbb{S}^2\)-bundle over \(\mathbb{S}^2\), branched over an embedded surface;NEWLINENEWLINENEWLINE\(\bullet\) if the collection of vanishing cycles for the fibration includes \(\sigma\) separating curves, then \(M\) is the relative minimalization of a Lefschetz fibration constructed as a 2-fold cover of \(\mathbb{C}\mathbb{P}^2\#(2\sigma+ 1)\overline{\mathbb{C}\mathbb{P}^2}\), branched over an embedded surface.NEWLINENEWLINENEWLINENote that independent proofs of the same results already exist: see [\textit{B. Siebert} and \textit{G. Tian}, Commun. Contemp. Math. 1, No. 2, 255-280 (1999; Zbl 0948.57018)] for the general case of hyperelliptic genus \(h\) Lefschetz fibrations, and [\textit{I. Smith}, Geom. Topol. 3, 211-233 (1999; Zbl 0929.53047)] for the particular case of genus two Lefschetz fibrations. However, the method of proof used by the author (which generalizes \textit{U. Persson's} work [Compos. Math. 43, 3-58 (1981; Zbl 0479.14018)]) is very interesting, since branched covers are fashioned by hand, beginning with handlebody descriptions of the involved manifolds.