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The simplification of singularities of Lagrangian and Legendrian fronts

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Abstract

We establish a full h-principle (\(C^0\)-close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy.

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References

  1. Alvarez-Gavela, D.: Refinements of the holonomic approximation lemma. Algebr. Geom. Topol. 18, 2265–2303 (2018)

    Article  MathSciNet  Google Scholar 

  2. Arnold, V.I.: A characteristic class entering in quantization conditions. Funct. Anal. Appl. 1(1), 1–13 (1967)

    Article  Google Scholar 

  3. Arnold, V.I.: Singularities of Caustics and Wave Fronts. Kluwer Academic Publishers, Dordrecht (1990)

    Book  Google Scholar 

  4. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Springer, Berlin (1985)

    Book  Google Scholar 

  5. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. II. Springer, Berlin (1988)

    Book  Google Scholar 

  6. Boardman, J.M.: Singularities of differentiable maps. Publ. Mathématiques de l’I.H.É.S 33(2), 21–57 (1967)

    Article  MathSciNet  Google Scholar 

  7. Ekholm, T.: Morse flow trees and Legendrian contact homology in 1-jet spaces. Geom. Topol. 11, 1083–1224 (2007)

    Article  MathSciNet  Google Scholar 

  8. Eliashberg, Y.M.: On singularities of folding type. Izv. Akad. Nauk SSSR Ser. Mat. 4(5), 1119–1134 (1970)

    MathSciNet  Google Scholar 

  9. Eliashberg, Y.M.: Surgery of singularities of smooth mappings. Izv. Akad. Nauk SSSR Ser. Mat. 36(6), 1321–1347 (1972)

    MathSciNet  Google Scholar 

  10. Eliashberg, Y.M., Mishachev, N.M.: Wrinkling of smooth mappings and its applications. I. Invent. Math. 130(2), 345–369 (1997)

    Article  MathSciNet  Google Scholar 

  11. Eliashberg, Y.M., Mishachev, N.M.: Wrinkling of smooth mappings. III. Foliations of codimension greater than one. Topol. Methods Nonlinear Anal. 11(2), 321–350 (1998)

    Article  MathSciNet  Google Scholar 

  12. Eliashberg, Y.M., Mishachev, N.M.: Wrinkling of smooth mappings. II. Wrinkling of embeddings and K. Igusa’s theorem. Topology 39(4), 711–732 (2000)

    Article  MathSciNet  Google Scholar 

  13. Eliashberg, Y.M., Mishachev, N.M.: Holonomic approximation and Gromov’s h-principle. arXiv:math/0101196

  14. Eliashberg, Y.M., Mishachev, N.M.: Introduction to the h-principle. Graduate Studies in Mathematics, vol. 48. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  15. Eliashberg, Y.M., Mishachev, N.M.: Wrinkled embeddings, foliations, geometry, and topology. Contemp. Math. 498, 207–232 (2009)

    Article  Google Scholar 

  16. Eliashberg, Y.M., Mishachev, N.M.: The space of framed functions is contractible. In: Pardalos, P., Rassias, T. (eds.) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg (2012)

    Google Scholar 

  17. Eliashberg, Y.M., Mishachev, N.M.: Topology of spaces of S-immersions. In: Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, vol. 296, Birkhäuser, Boston, MA (2012)

  18. Eliashberg, Y.M., Galatius, S., Mishachev, N.M.: Madsen–Weiss for geometrically minded topologists. Geom. Topol. 15(1), 411–472 (2011)

    Article  MathSciNet  Google Scholar 

  19. Eliashberg, Y.M.: Weinstein manifolds revisited, arXiv:1707.03442

  20. Entov, M.: Surgery on Lagrangian and Legendrian singularities. Geom. Funct. Anal. 9(2), 298–352 (1999)

    Article  MathSciNet  Google Scholar 

  21. Entov, M.: On the necessity of Legendrian fold singularities. IMRN International Mathematics Research Notices, No. 20 (1998)

  22. Emmanuel, F., Pushkar’, E.P.: Non cancellation of singularities on wave fronts. C. R. Acad. Sci. Paris. Ser. I Math. 327(8), 827–831 (1998)

    MATH  Google Scholar 

  23. Emmanuel, F., Pushkar’, E.P.: Morse theory and global coexistence of singularities on wave fronts. J. Lond. Math. Soc. 74(2), 527–544 (2006)

    Article  MathSciNet  Google Scholar 

  24. Gromov, M.L.: Stable maps of foliations into manifolds. Izv. Akad. Nauk SSSR Ser. Mat. 33(4), 671 (1969)

    MATH  Google Scholar 

  25. Gromov, M.L.: Convex integration of partial differential relations. Izv. Akad. Nauk SSSR Ser. Mat. 37, 329–343 (1973)

    MathSciNet  MATH  Google Scholar 

  26. Gromov, M.L.: Partial Differential Relations. Springer, Berlin (1986)

    Book  Google Scholar 

  27. Hirsch, M.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)

    Article  MathSciNet  Google Scholar 

  28. Igusa, K.: Higher singularities are unnecessary. Ann. Math. 119, 1–58 (1984)

    Article  MathSciNet  Google Scholar 

  29. Igusa, K.: The space of framed functions. Trans. Am. Math. Soc. 301(2), 431–477 (1987)

    Article  MathSciNet  Google Scholar 

  30. Kragh, T.: Parametrized ring-spectra and the nearby Lagrangian conjecture, arXiv:1107.4674 (2011)

  31. Kupers, S.: Three applications of delooping applied to H-principles, arXiv:1701.06788

  32. Lurie, J.: On the classification of topological field theories, arXiv:0905.0465

  33. Morin, B.: Formes canoniques des singularites d’une application differetiable. C. R. Acad. Sci. Paris 260, 5662–5665 (1965)

    MathSciNet  MATH  Google Scholar 

  34. Murphy, E.: Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245 (2012)

  35. Nadler, D.: Arboreal singularities, arXiv:1309.4122

  36. Nadler, D.: Non-characteristic expansions of Lagrangian singularities, arxiv:1507.01513

  37. Phillips, A.: Submersions of open manifolds. Topology 6, 171–206 (1967)

    Article  MathSciNet  Google Scholar 

  38. Rourke, C., Sanderson, B.: The compression theorem I–II. Geom. Topol. 5, 399–429 (2001). (431–440)

    Article  MathSciNet  Google Scholar 

  39. Rourke, C., Sanderson, B.: The compression theorem III: applications. Algebr. Geom. Topol. 3, 857–872 (2003)

    Article  MathSciNet  Google Scholar 

  40. Smale, S.: The classification of immersions of spheres in Euclidean spaces. Ann. Math. 2(69), 327–344 (1959)

    Article  MathSciNet  Google Scholar 

  41. Spring, D.: Convex Integration Theory. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  42. Spring, D.: Directed embeddings and the simplification of singularities. Commun. Contemp. Math. 4, 107–144 (2002)

    Article  MathSciNet  Google Scholar 

  43. Spring, D.: Directed embeddings of closed manifolds. Commun. Contemp. Math. 7, 707–725 (2005)

    Article  MathSciNet  Google Scholar 

  44. Starkston, L.: Arboreal singularities in Weinstein Skeleta, arXiv:1707.03446

  45. Thom, R.: Les singularités des applications différentiables. Annales de l’Institut Fourier 6(6), 43–87 (1956)

    Article  MathSciNet  Google Scholar 

  46. Thurston, W.: The theory of foliations of codimension greater than one. Commentarii Mathematici Helvetici 49, 214–231 (1974)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I am very grateful to my advisor Yasha Eliashberg for insightful guidance throughout this project. I would also like to thank Laura Starkston for reading carefully the first draft of this paper and offering numerous remarks and corrections which have greatly improved the exposition. I am indebted to the ANR Microlocal group who held a workshop in January 2017 to dissect an early version of the paper and in particular to Sylvain Courte and Alexandre Vérine who spotted several mistakes in the proof of the local wrinkling lemma and made useful suggestions for fixing them. My gratitude also goes to Roger Casals, Sander Kupers, Emmy Murphy, Oleg Lazarev and Kyler Siegel for many helpful discussions surrounding the general notion of flexibility. Finally, many thanks to the referee for numerous helpful comments and corrections.

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  1. Stanford University, Stanford, USA

    Daniel Álvarez-Gavela

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  1. Daniel Álvarez-Gavela
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Correspondence to Daniel Álvarez-Gavela.

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The author was partially supported by NSF Grant DMS-1505910.

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Álvarez-Gavela, D. The simplification of singularities of Lagrangian and Legendrian fronts. Invent. math. 214, 641–737 (2018). https://doi.org/10.1007/s00222-018-0811-3

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