Volume 20, issue 7 (2020)

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Fundamental groups of formal Legendrian and horizontal embedding spaces

Eduardo Fernández, Javier Martínez-Aguinaga and Francisco Presas

Algebraic & Geometric Topology 20 (2020) 3219–3312
Abstract

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in 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 4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at π1–level.

Keywords
Legendrian knots, horizontal knots, formal embeddings
Mathematical Subject Classification
Primary: 57R17, 58A17
Secondary: 53D10
References
Publication
Received: 27 November 2017
Revised: 11 November 2019
Accepted: 9 December 2019
Published: 29 December 2020
Authors
Eduardo Fernández
Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, Universidad Complutense de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Madrid
Spain
Javier Martínez-Aguinaga
Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, Universidad Complutense de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Madrid
Spain
Francisco Presas
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Madrid
Spain