Iterated fibre sums of algebraic Lefschetz fibrations (Q2922460)

In [4-manifolds and Kirby calculus. Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0933.57020)], \textit{R. E. Gompf} and \textit{A. I. Stipsicz} showed that if \(E(1)\) is the elliptic surface, then by using fiber summing along general fibers the sequence of simply connected elliptic surfaces \(E(n)\) without multiple fibers can be constructed.NEWLINENEWLINEIn this paper, the author generalizes the construction of the surfaces \(E(n)\) by considering the \(n\)-fold fiber sum \(M(n)\). If \(M'\) is an arbitrary smooth algebraic surface, then it admits a Lefschetz pencil which extends to a Lefschetz fibration on some blowup \(M\). By considering the iterated fiber sums of these fibrations, a sequence \(M(n)\) of symplectic manifolds with an induced Lefschetz fibration over \(\mathbb{CP}^1\) is obtained. The author describes the basic topology of these manifolds. It is shown that if \(M\) is simply connected, then all \(M(n)\) are simply connected and there is a description of the intersection form and of the canonical class. Also, if \(M'\) is a minimal surface of general type, then the only basic class up to sign of \(M(n)\) for all \(n\geq 2\) which has non-zero intersection with the fiber is the canonical class.NEWLINENEWLINEGompf and Stipsicz, in the above mentioned book, showed that for \(E(1)\) every diffeomorphism on \(\partial\nu\Sigma\) extends over \(E(1)\setminus \text{int\,}\nu\Sigma\) and for \(E(n)\) only if it preserves the torus fibration on the boundary. In this paper, the author determines which orientation-preserving self-diffeomorphisms of the boundary of the tubular neighborhood \(\nu\Sigma\) of a general fiber in \(M(n)\) extend over the complement \(M(n)\setminus \text{int\,}\nu\Sigma\), and derives a criterion that can be used to show that certain diffeomorphisms do not extend over the complement.