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Realizing 4-manifolds as achiral Lefschetz fibrations
We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, if X is a simply connected 4-manifold we show that X#S 2 × S 2 and X#S 2×eS 2 both admit achiral Lefschetz fibrations. We also show 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.