Realizing 4-manifolds as achiral Lefschetz fibrations

Abstract

We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, if <it>X</it> is a simply connected 4-manifold, we show that <it>X</it>#<it>S</it>2×<it>S</it>2 and $$X\\#{S}^{2}\\tilde{\\times }{S}^{2}$$ both admit achiral Lefschetz fibrations. We also show that these surgered manifolds admit near-symplectic structures, and prove more generally that achiral Lefschetz fibrations with sections have near-symplectic structures. As a corollary to our proof, we obtain an alternate proof of Gompf's result on the existence of symplectic structures on Lefschetz pencils.