Geometry & Topology

The $h$–principle for broken Lefschetz fibrations

Jonathan Williams

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Abstract

It is known that an arbitrary smooth, oriented 4–manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken Lefschetz fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken Lefschetz fibration in a given homotopy class of smooth maps. One notable application is that adding an additional “projection" move generates all broken Lefschetz fibrations, regardless of homotopy class. The paper ends with further applications and open problems.

Article information

Source
Geom. Topol., Volume 14, Number 2 (2010), 1015-1061.

Dates
Received: 1 July 2009
Accepted: 23 February 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732211

Digital Object Identifier
doi:10.2140/gt.2010.14.1015

Mathematical Reviews number (MathSciNet)
MR2629899

Zentralblatt MATH identifier
1204.57027

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]
Secondary: 57R70: Critical points and critical submanifolds 57R17: Symplectic and contact topology

Keywords
broken Lefschetz fibration $4$–manifold stable map

Citation

Williams, Jonathan. The $h$–principle for broken Lefschetz fibrations. Geom. Topol. 14 (2010), no. 2, 1015--1061. doi:10.2140/gt.2010.14.1015. https://projecteuclid.org/euclid.gt/1513732211


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References

  • S Akbulut, Ç Karakurt, Every $4$–manifold is BLF, J. Gökova Geom. Topol. GGT 2 (2008) 83–106
  • Y Ando, On the elimination of Morin singularities, J. Math. Soc. Japan 37 (1985) 471–487
  • Y Ando, On local structures of the singularities $A\sb k\;D\sb k$ and $E\sb k$ of smooth maps, Trans. Amer. Math. Soc. 331 (1992) 639–651
  • V I Arnold, Catastrophe theory, third edition, Springer, Berlin (1992) Translated from the Russian by G S Wassermann, Based on a translation by R K Thomas
  • V I Arnold, S M Guseĭn-Zade, A N Varchenko, Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts, Monogr. in Math. 82, Birkhäuser, Boston (1985) Translated from the Russian by I Porteous and M Reynolds
  • D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043–1114
  • R \I Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. (2008) Art. ID rnn 101, 15pp
  • R \I Baykur, Handlebody argument for modifying achiral singularities (appendix to [L1)?], Geom. Topol. 13 (2009) 312–317
  • R \I Baykur, Topology of broken Lefschetz fibrations and near-symplectic four-manifolds, Pacific J. Math. 240 (2009) 201–230
  • S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743–785
  • Y Eliashberg, N M Mishachev, Wrinkling of smooth mappings and its applications. I, Invent. Math. 130 (1997) 345–369
  • Y Eliashberg, N Mishachev, Introduction to the $h$–principle, Graduate Studies in Math. 48, Amer. Math. Soc. (2002)
  • D T Gay, R Kirby, Constructing Lefschetz-type fibrations on four-manifolds, Geom. Topol. 11 (2007) 2075–2115
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc. (1999)
  • Y Lekili, Heegaard Floer homology of broken fibrations over the circle, Preprint (2009)
  • Y Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. 13 (2009) 277–318 Appendix B by R \.I Baykur
  • H I Levine, Elimination of cusps, Topology 3 (1965) 263–296
  • P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326–400
  • T Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007) 759–828
  • O Saeki, Elimination of definite fold, Kyushu J. Math. 60 (2006) 363–382
  • C H Taubes, Counting pseudo-holomorphic submanifolds in dimension $4$, J. Differential Geom. 44 (1996) 818–893
  • M Usher, The Gromov invariant and the Donaldson–Smith standard surface count, Geom. Topol. 8 (2004) 565–610
  • G Wassermann, Stability of unfoldings in space and time, Acta Math. 135 (1975) 57–128