Diffeomorphism types of genus 2 Lefschetz fibrations (Q1392401)

A genus \(g\) Lefschetz fibration is a smooth mapping \(f:M\to \mathbb{S}^2\) \((M\) being a smooth oriented 4-manifold) so that the regular fiber is diffeomorphic to an oriented genus \(g\) surface \(\Sigma_g\), and \(f\) is locally given -- in suitable complex coordinate neighborhoods of each critical point of \(M\) -- by \(f(z,w) =z^2 +w^2\). According to \textit{A. Kas} [Pac. J. Math. 89, 89-104 (1980; Zbl 0457.14011)], for \(g\geq 2\) every such fibration is uniquely determined by a trivial element in the mapping class group of \(\Sigma_g\). The present paper takes into account, in the case \(g=2\), the genus two Lefschetz fibrations \(f_1:M_1 \to\mathbb{S}^2\) and \(f_2: M_2\to \mathbb{S}^2\) induced by two known relations in the mapping class group of \(\Sigma_2\) [\textit{J. S. Birman}, Braids, links, and mapping class groups, Ann. Math. Stud. 82 (1975; Zbl 0305.57013)] and \textit{Y. Matsumoto} [in: Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24--28, 1995. Singapore: World Scientific. 123--148 (1996; Zbl 0921.57006)], where \(M_1\) and \(M_2\) were proved to be both homeomorphic to \(5\mathbb{C} \mathbb{P}^2 \neq \overline {\mathbb{C}\mathbb{P}^2})\). By means of double branched covering techniques, the smooth 4-manifolds \(M_1\) and \(M_2\) are now entirely identified as complex surfaces; as a consequence, it is proved that \(M_1\), \(M_2\) and \(5\mathbb{C} \mathbb{P}^2 \neq\overline {\mathbb{C} \mathbb{P}^2}\) are mutually non-diffeomorphic. This provides a complete negative answer to a question contained in the Kirby problem list [\textit{Kirby}, Problems in low-dimensional topology (in: Geometric topology, 1993 Georgia international topology conference (1997; Zbl 0882.00041)]) and originally raised by Matsumoto in the quoted paper.