Geometry & Topology

Lagrangian matching invariants for fibred four-manifolds: I

Tim Perutz

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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

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

Received: 7 June 2006
Revised: 20 April 2007
Accepted: 27 March 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D40: Floer homology and cohomology, symplectic aspects 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)

Four-manifolds Lefschetz fibrations Seiberg–Witten invariants pseudo-holomorphic curves Lagrangian submanifolds Hilbert schemes


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.

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  • D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043–1114
  • L Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994) 589–660
  • S K Donaldson, Geometry of four-manifolds, ICM Series, Amer. Math. Soc. (1988) A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986, Introduced by I M James
  • S K Donaldson, Topological field theories and formulae of Casson and Meng–Taubes, from: “Proceedings of the Kirbyfest (Berkeley, CA, 1998)”, (M Scharlemann, J Hass, editors), Geom. Topol. Monogr. 2 (1999) 87–102
  • S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743–785
  • J J Duistermaat, G J Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259–268
  • D Gay, R Kirby, Constructing Lefschetz-type fibrations on four–manifolds
  • D Gieseker, A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984) 173–206
  • V Guillemin, E Lerman, S Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press (1996)
  • A Hatcher, On the diffeomorphism group of $S\sp{1}\times S\sp{2}$, Proc. Amer. Math. Soc. 83 (1981) 427–430
  • P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. $(2)$ 165 (2007) 457–546
  • Y-J Lee, Heegaard splittings and Seiberg-Witten monopoles, from: “Geometry and topology of manifolds”, Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 173–202
  • I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319–343
  • D McDuff, D Salamon, Introduction to symplectic topology, second edition, Oxford Mathematical Monographs, Oxford University Press (1998)
  • D S Nagaraj, C S Seshadri, Degenerations of the moduli spaces of vector bundles on curves. I, Proc. Indian Acad. Sci. Math. Sci. 107 (1997) 101–137
  • H Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series 18, Amer. Math. Soc. (1999)
  • R Pandharipande, A compactification over $\overline {M}\sb g$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996) 425–471
  • T Perutz, Surface–fibrations, four–manifolds, and symplectic Floer homology, PhD thesis, Imperial College, London (2005)
  • T Perutz, Lagrangian matching invariants for fibred four–manifolds: II
  • Z Ran, A note on Hilbert schemes of nodal curves, J. Algebra 292 (2005) 429–446
  • W-D Ruan, Deformation of integral coisotropic submanifolds in symplectic manifolds, J. Symplectic Geom. 3 (2005) 161–169
  • D A Salamon, Spin geometry and Seiberg–Witten invariants, unpublished book
  • D A Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers, from: “Proceedings of 6th Gökova Geometry-Topology Conference”, Turkish J. Math. 23 (1999) 117–143
  • P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003–1063
  • P Seidel, I Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006) 453–514
  • I Smith, Serre-Taubes duality for pseudoholomorphic curves, Topology 42 (2003) 931–979
  • C H Taubes, The geometry of the Seiberg-Witten invariants, from: “Surveys in differential geometry, Vol. III (Cambridge, MA, 1996)”, Int. Press, Boston (1998) 299–339
  • C H Taubes, Seiberg Witten and Gromov invariants for symplectic $4$-manifolds, First International Press Lecture Series 2, International Press, Somerville, MA (2000) Edited by Richard Wentworth
  • M Usher, The Gromov invariant and the Donaldson-Smith standard surface count, Geom. Topol. 8 (2004) 565–610
  • M Usher, Vortices and a TQFT for Lefschetz fibrations on 4-manifolds, Algebr. Geom. Topol. 6 (2006) 1677–1743