On the number of vanishing cycles in Lefschetz fibrations (Q1574729)

The main result of the paper is an lower bound for the number of homologically nontrivial vanishing cycles of a genus-\(g\) Lefschetz fibration. A Lefschetz fibration is any map of a 4-manifold onto the 2-sphere with only a finite number of singular points having a local model \(z^2_1+z^2_2\) at each of them. If the fibration is nontrivial then the number of homologically nontrivial vanishing cycles (and hence of singular fibers) is bounded from below by \(1/5(4g +2)\), where \(g\) is the genus of the nonsingular fiber.