In a pair of papers, we construct invariants for smooth four-manifolds equipped with 'broken fibrations'-the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov-generalising the Donaldson-Smith invariants for Lefschetz fibrations.The 'Lagrangian matching invariants' are designed to be comparable with the SeibergWitten invariants of the underlying four-manifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to three-manifolds fibred over S 1 .The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I-the present paper-is devoted to the symplectic geometry of these Lagrangians.53D40, 57R57; 57R15
We give elementary proofs of two 'folklore' assertions about near-symplectic forms on four-manifolds: that any such form can be modified, by an evolutionary process taking place inside a finite set of balls, so as to have any chosen positive number of zero-circles; and that, on a closed manifold, the number of zero-circles for which the splitting of the normal bundle is trivial has the same parity as 1 + b1 + b + 2 .
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