A remark on the contactomorphism group of overtwisted contact spheres

TL;DR

This paper demonstrates the existence of infinite order elements in the homotopy groups of contactomorphism groups of overtwisted spheres, revealing their complex topological structure unlike finite-dimensional Lie groups.

Contribution

It establishes the presence of infinite order elements in the homotopy groups of contactomorphism groups of overtwisted spheres, highlighting their non-Lie group nature.

Findings

Existence of infinite order elements in homotopy groups

Contactomorphism groups are not homotopically equivalent to finite-dimensional Lie groups

High-dimensional overtwisted spheres have complex contactomorphism groups

Abstract

We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some high dimensional overtwisted spheres is not homotopically equivalent to a finite dimensional Lie group.