Existence h-principle for Engel structures


In this article we prove that the inclusion of the space of Engel structures of a smooth 4-manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. In particular, we construct Engel structures representing any given full flag.

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We are grateful to V. Colin, V. L. Ginzburg, E. Giroux, E. Murphy, K. Niederkrüger, L. E. Solá-Conde, and A. Stipsicz for useful discussions. We would like to especially acknowledge Y. Eliashberg and T. Vogel for intense and valuable discussions during the conference h-Principles in Houat; the arguments in this article have been greatly simplified thanks to them. The classical construction explained in Example 9 was pointed out to us by Daniel Fox and it has been an important intuition for the development of this work. Thanks as well to the anonymous referee for useful comments and suggestions. The authors are supported by Spanish National Research Project MTM2013—42135. This work is supported in part by the ICMAT Severo Ochoa grant SEV-2011-0087 through the V. Ginzburg Lab, Á. del Pino is supported by La Caixa–Severo Ochoa grant and J. L. Pérez is supported by a MINECO FPI grant.

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Correspondence to Roger Casals.

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Casals, R., Pérez, J.L., del Pino, Á. et al. Existence h-principle for Engel structures. Invent. math. 210, 417–451 (2017). https://doi.org/10.1007/s00222-017-0732-6

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Mathematics Subject Classification

  • Primary 53A40
  • 53D35
  • 58A17