Lagrangian matching invariants for fibred four-manifolds: I

Details Lagrangian matching invariants for fibred four-manifolds: I Lagrangian matching invariants for fibred four-manifolds: I

2006-06-02

arxiv.org/abs/math/0606061v2

Geom. Topol. 11 (2007), 759--828

Mathematics

Symplectic Geometry

72 pages, 4 figures. v.2 - numerous small corrections and clarifications

Scientific paper

10.2140/gt.2007.11.759

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. They fit into a field theory which assigns Floer homology groups to fibred 3-manifolds. 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 is devoted to the symplectic geometry of these Lagrangians.

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