## Geometry & Topology

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

Tim Perutz

#### 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 $S1$.

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.

#### Article information

Source
Geom. Topol., Volume 11, Number 2 (2007), 759-828.

Dates
Revised: 20 April 2007
Accepted: 27 March 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799861

Digital Object Identifier
doi:10.2140/gt.2007.11.759

Mathematical Reviews number (MathSciNet)
MR2302502

Zentralblatt MATH identifier
1143.53079

#### Citation

Perutz, Tim. Lagrangian matching invariants for fibred four-manifolds: I. Geom. Topol. 11 (2007), no. 2, 759--828. doi:10.2140/gt.2007.11.759. https://projecteuclid.org/euclid.gt/1513799861

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