Volume 11, issue 2 (2007)

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Lagrangian matching invariants for fibred four-manifolds: I

Tim Perutz

Geometry & Topology 11 (2007) 759–828
 arXiv: math.SG/0606061
Abstract

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 Seiberg–Witten 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.

Keywords
Four-manifolds, Lefschetz fibrations, Seiberg–Witten invariants, pseudo-holomorphic curves, Lagrangian submanifolds, Hilbert schemes
Mathematical Subject Classification 2000
Primary: 53D40, 57R57
Secondary: 57R15