On four-manifolds fibering over surfaces (Q1277151)

This is a paper regarding topological and smooth 4-manifolds that fiber over surfaces. In [\textit{J. A. Hillman}, Topology Appl. 40, No. 3, 275-286 (1991; Zbl 0754.57014); The algebraic characterization of geometric 4-manifolds, Lond. Math. Soc. Lect. Note Ser. 198 (1994; Zbl 0812.57001)] necessary and sufficient conditions for a 4-manifold to have the homotopy type of a surface bundle over a surface are given. A 4-manifold \(X\) is a homotopy surface bundle if and only if \(\chi(X) = \chi(F_1) \chi(F_2)\) and \(\pi_1 (X)\) is an extension of \(\pi_1 (F_1)\) by \(\pi_1 (F_2)\). This paper is aimed at the question of how many fiber structures exist for a given 4-manifold and pair of surfaces. The paper begins with an analysis of the classifying space of the mapping class group. We should remark that the cohomology class of this classifying space corresponding to the signature of the 4-manifold is related to Casson's invariant. After the analysis of this classifying space, the authors prove that the spectral sequence of the fibration collapses. One interesting application of this result is that Rokhlin's theorem holds for this class of 4-manifolds (even the topological 4-manifolds).