A stable classification of Lefschetz fibrations

Abstract

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a “universal” fibration $f_g^0$ with the property that, if two Lefschetz fibrations over $S^2$ have the same Euler–Poincaré characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with $f_g^0$ they become isomorphic. As a consequence, any two compact integral symplectic 4–manifolds with the same values of $\left(c_1^2,c_2,c_1\cdot \left[\omega \right],{\left[\omega \right]}^2\right)$ become symplectomorphic after blowups and symplectic sums with $f_g^0$.