On the contact mapping class group of Legendrian circle bundles

TL;DR

This paper characterizes the contact transformation group of Legendrian circle bundles over surfaces, extending previous results and emphasizing the role of embedding space connectedness in contact topology.

Contribution

It provides a complete description of the contact mapping class group for Legendrian circle bundles, correcting earlier work and exploring embedding space connectedness.

Findings

Determined the contact transformation group for Legendrian circle bundles.

Extended and corrected previous results on contact mapping class groups.

Analyzed the connectedness of embedding spaces in contact 3-manifolds.

Abstract

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.