Volume 11, issue 2 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 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.

Keywords
Four-manifolds, Lefschetz fibrations, Seiberg–Witten invariants, pseudo-holomorphic curves, Lagrangian submanifolds, Hilbert schemes
Mathematical Subject Classification 2000
Primary: 53D40, 57R57
Secondary: 57R15
References
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Publication
Received: 7 June 2006
Revised: 20 April 2007
Accepted: 27 March 2007
Published: 10 May 2007
Proposed: Peter Ozsváth
Seconded: Simon Donaldson, Ron Fintushel
Authors
Tim Perutz
DPMMS
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
UK
http://www.dpmms.cam.ac.uk/~tp214