On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

TL;DR

This paper investigates the topological structure of spaces of loops of contactomorphisms and Legendrian submanifolds in non-orderable contact manifolds, revealing relationships between their homotopy groups and positivity filtrations.

Contribution

It introduces a filtration based on positivity to analyze the topology of loop spaces and establishes subgroup relations between their homotopy groups.

Findings

Homotopy groups of loop spaces are subgroups of those of positive loops.

Topology of loop spaces is described via a positivity filtration.

Analogous results are obtained for Legendrian loop spaces.

Abstract

We study the global topology of the space $\mathcal L$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M^{2n+1}, \alpha)$. We filter $\mathcal L$ by a quantitative measure of the ``positivity'' of the loops and describe the topology of $\mathcal L$ in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of $\mathcal L$ are subgroups of the homotopy groups of the subspace of positive loops $\mathcal L^+$. We obtain analogous results for the space of loops of Legendrian submanifolds in $(M^{2n+1}, \alpha)$.