Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at π-level.
The authors are supported by Spanish Research Projects SEV?2015?0554, MTM2016?79400?P and MTM2015?72876?EXP. Fernández is supported by a Master?Severo Ochoa grant and by Beca de Personal Investigador en Formación UCM. MartínezAguinaga is funded by Programa Predoctoral de Formación de Personal Investigador No Doctor del Departamento de Educación del Gobierno Vasco.