On the classification of prolongations up to Engel homotopy
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Abstract:
In (Casals, Pérez, del Pino, and Presas, preprint) it was shown that Engel structures satisfy an existence $h$–principle, and the question of whether a full $h$–principle holds was left open. In this note we address the classification problem, up to Engel deformation, of Cartan and Lorentz prolongations. We show that it reduces to their formal data as soon as the turning number is large enough.
Somewhat separately, we study the homotopy type of the space of Cartan prolongations, describing completely its connected components in the overtwisted case.
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Additional Information
- Álvaro del Pino
- Affiliation: Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas – CSIC, C. Nicolás Cabrera, 13–15, 28049, Madrid, Spain
- Email: alvaro.delpino@icmat.es
- Received by editor(s): October 29, 2016
- Received by editor(s) in revised form: February 15, 2017, and March 16, 2017
- Published electronically: August 1, 2017
- Additional Notes: The author was supported by the Spanish Research Projects SEV–2015–0554, MTM2013–42135, and MTM2015–72876–EXP and a La Caixa–Severo Ochoa grant
- Communicated by: Ken Bromberg
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 891-907
- MSC (2010): Primary 58A30
- DOI: https://doi.org/10.1090/proc/13751
- MathSciNet review: 3731719