The topology of a subspace of the Legendrian curves in a closed contact 3-manifold

TL;DR

This paper investigates a specific subspace of Legendrian loops in a closed contact 3-manifold, demonstrating its homotopy equivalence to the full loop space and highlighting its role in contact form geometry.

Contribution

It establishes that the subspace of zero Maslov index Legendrian loops is homotopically equivalent to the entire Legendrian loop space, providing new insights into contact geometry variations.

Findings

The subspace is S1-equivariantly homotopy equivalent to the full loop space.

The subspace corresponds to Legendrian loops with zero Maslov index.

This space serves as a natural setting for variations in contact form geometry.

Abstract

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.