Diffeomorphism types of good torus fibrations with twin singular fibers (Q2780728)

A good torus fibration (GTF), originally defined by Y. Matsumoto in the 1970's, is a smooth map \(f\) from a smooth, oriented 4-manifold \(M\) to a smooth, oriented 2-manifold \(B\) such that \(f\) describes \(M\) as a smooth torus bundle over \(B-\Gamma\), where \(\Gamma\) is a set of isolated points in the interior of \(B\); furthermore, and most importantly, the singular fibers \(f^{-1}(x)\) for \(x\in\Gamma\) are assumed to have at most a finite number of normal crossings. The simplest such singular fibers consist of an immersed sphere with a single transverse self-intersection, and are denoted \(I_1^+\) or \(I_1^-\) according to the sign of the self-intersection. Also of interest are twin singular fibers, which consist of two embedded spheres intersecting in two points with opposite signs. \textit{Y. Matsumoto} proved [J. Math. Soc. Japan 37, 605-636 (1985; Zbl 0624.57017); Topology 25, 549-563 (1986; Zbl 0615.14023)] that the diffeomorphism type of any \(M\) admitting a GTF with singular fibers only of type \(I_1^{\pm}\) and with signature \(\sigma(M)\neq 0\) is determined by the Euler characteristic \(e(M)\) of \(M\), \(\sigma(M)\), and the genus \(g(B)\) of \(B\). \textit{Z. Iwase} [Jap. J. Math., New Ser. 10, 321-352 (1984; Zbl 0603.57014)] considered the case where \(\sigma(M)=0\), showing in this case that if \(M\) admits a GTF over \(S^2\) with only non-multiple twin singular fibers, then \(M\) is determined by \(e(M)\), \(\pi_1(M)\), whether or not \(M\) is spin, and the parity of the intersection form on \(H_2(M)\). NEWLINENEWLINENEWLINEThe paper under review treats the remaining case where \(\sigma(M)=0\) and the base \(B\) is an arbitrary oriented surface. The author shows that if \(M\) admits a GTF with only non-multiple twin singular fibers, at least one even twin singular fiber (a condition defined within), and \(\operatorname {rank} H_1(M)\) odd, then the diffeomorphism type of \(M\) is determined by \(e(M)\), \(g(B)\), and the parity of the intersection form on \(H_2(M)\). In addition, he shows that if each singular fiber of a GTF is of type \(I_1^{\pm}\), \(e(M)\neq 0\), and \(\operatorname {rank} H_1(M)\) is odd, then \(M\) is determined by \(e(M)\) and \(g(B)\). Finally, addressing the case where \(\operatorname {rank} H_1(M)\) is even, the author determines the diffeomorphism type of GTF's with \(e(M)=2\), \(g(B)=1\), each singular fiber a non-multiple even twin singular fiber, and \(\pi_1(M)\) in certain prescribed classes.