Fundamental groups of formal Legendrian and horizontal embedding spaces

TL;DR

This paper calculates the fundamental groups of spaces of formal Legendrian and horizontal embeddings in R3 and R4, revealing that some non-trivial loops are already non-trivial at the formal level, thus advancing understanding of their topological structure.

Contribution

It provides the first explicit computation of the fundamental groups of these embedding spaces, connecting formal and actual embedding properties.

Findings

Non-trivial loops in Legendrian embeddings are already non-trivial at the formal level.

Computed fundamental groups of formal horizontal embeddings in R4.

Confirmed that these formal computations match the fundamental groups of actual embedding spaces.

Abstract

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 non trivial loops of Legendrian embeddings are already non trivial 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 this computes the fundamental group of the space of horizontal embeddings as well.