Fundamental groups of formal Legendrian and horizontal embedding spaces

Abstract

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in ${ℝ}^3$. 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 ${ℝ}^4$. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at ${\pi }_1$–level.