Skip to main content
Log in

Existence of two-parameter crossings, with applications

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A Morse 2-function is a generic smooth map f from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy of f and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the theory of crown diagrams of smooth 4-manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Auroux, D., Donaldson, S., Katzarkov, L.: Singular Lefschetz pencils. Geom. Topol. 9, 1043–1114 (2005). https://doi.org/10.2140/gt.2005.9.1043

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbulut, S., Karakurt, Ç.: Every 4-manifold is BLF. J. Gökova Geom. Topol. 2, 40–82 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Baykur, R.I.: Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds. Pac. J. Math. 240(2), 201–230 (2009). https://doi.org/10.2140/pjm.2009.240.201

    Article  MathSciNet  MATH  Google Scholar 

  4. Baykur, R.I.: Existence of broken Lefschetz fibrations. Int. Math. Res. Not. (2008). https://doi.org/10.1093/imrn/rnn101

    Article  MATH  Google Scholar 

  5. Baykur, R.I., Saeki, O.: Simplifying indefinite fibrations on 4-manifolds, preprint (2017)

  6. Behrens, S.: On 4-manifolds, folds and cusps. Pac. J. Math. 264(2), 257–306 (2013). https://doi.org/10.2140/pjm.2013.264.257

    Article  MathSciNet  MATH  Google Scholar 

  7. Behrens, S., Hayano, K.: Elimination of cusps in dimension 4 and its applications. Proc. Lond. Math. Soc. 113(5), 674–724 (2016). https://doi.org/10.1112/plms/pdw042

    Article  MathSciNet  MATH  Google Scholar 

  8. Èliashberg, Y.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98(3), 623–637 (1989)

    Article  MathSciNet  Google Scholar 

  9. Etnyre, J.: Lectures on open book decompositions and contact structures, from: “Floer homology, gauge theory, and low-dimensional topology. In: Ellwood, D., Ozsváth, P., Stipsicz, A., Szabó, Z. (eds.) Clay Mathematics Proceedings, vol. 5, pp. 103–141. American Mathematical Society (2006)

  10. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  11. Gay, D., Kirby, R.: Constructing Lefschetz-type fibrations on 4-manifolds. Geom. Topol. 11, 2075–2115 (2007). https://doi.org/10.2140/gt.2007.11.2075

    Article  MathSciNet  MATH  Google Scholar 

  12. Gay, D., Kirby, R.: Indefinite Morse 2-functions; broken fibrations and generalizations. Geom. Topol. 19, 2465–2534 (2015)

    Article  MathSciNet  Google Scholar 

  13. Hass, J., Thompson, A., Thurston, W.: Stabilization of Heegaard splittings. Geom. Topol. 13, 2029–2050 (2009). https://doi.org/10.2140/gt.2009.13.2029

    Article  MathSciNet  MATH  Google Scholar 

  14. Hayano, K.: Modification rule of monodromies in an \(R_2\)-move. Alg. Geom. Topol. 14, 2181–2222 (2014). https://doi.org/10.2140/agt.2014.14.2181

    Article  MathSciNet  MATH  Google Scholar 

  15. Laudenbach, F.: A proof of Reidemeister–Singer’s theorem by Cerf’s methods. Annales de la Faculté des sciences de Toulouse Mathématiques XXIII 1, 197–221 (2014). https://doi.org/10.5802/afst.1404

    Article  MathSciNet  MATH  Google Scholar 

  16. Lekili, Y.: Wrinkled fibrations on near-symplectic manifolds. Geom. Topol. 13, 277–318 (2009). https://doi.org/10.2140/gt.2009.13.277

    Article  MathSciNet  MATH  Google Scholar 

  17. Saeki, O.: Elimination of definite fold. Kyushu J. Math. 60, 363–382 (2006). https://doi.org/10.2206/kyushujm.60.363

    Article  MathSciNet  MATH  Google Scholar 

  18. Williams, J.: The \(h\)-principle for broken Lefschetz fibrations. Geom. Topol. 14(2), 1015–1061 (2010). https://doi.org/10.2140/gt.2010.14.1015

    Article  MathSciNet  MATH  Google Scholar 

  19. Williams, J.: Uniqueness of crown diagrams of smooth \(4\)-manifolds, preprint (2011)

Download references

Acknowledgements

The author would like to thank Denis Auroux, David Gay and Yankı Lekili for helpful conversations during the preparation of this work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jonathan D. Williams.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Williams, J.D. Existence of two-parameter crossings, with applications. Geom Dedicata 207, 265–286 (2020). https://doi.org/10.1007/s10711-019-00499-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00499-1

Keywords

Mathematics Subject Classification

Navigation