Four-manifolds admitting hyperelliptic broken Lefschetz fibrations (Q357524)

It was proved independently by \textit{T. Fuller} [Pac. J. Math. 196, No. 2, 369--393 (2000; Zbl 0971.57027)] and \textit{B. Siebert} and \textit{G. Tian} [Commun. Contemp. Math. 1, No. 2, 255--280 (1999; Zbl 0948.57018)] that, after blowing up \(s\) times, the total space of a hyperelliptic Lefschetz fibration (over \(S^2\)) is a double branched covering of \((S^2\times S^2)\# 2s\overline{\mathbb {CP}^2}\), where \(s\) is the number of separating Lefschetz singularities.NEWLINENEWLINEIn the paper under review, this result is extended to hyperelliptic simplified broken Lefschetz fibrations \(f: M\rightarrow S^2\) (introduced in [\textit{R. I. Baykur}, Pac. J. Math. 240, No. 2, 201--230 (2009; Zbl 1162.57011)]) with fiber genus \(g\geq 3\). This is proved through the construction of an involution on \(M\# s \overline{\mathbb {CP}^2}\). Moreover, it is shown that a regular fiber represents a nontrivial rational homology class of \(M\). As an immediate corollary, a closed oriented four-manifold with definite intersection form does not admit any simplified broken Lefschetz fibrations with fiber genus \(g\geq 3\).