A Remark on the Reeb Flow for Spheres

TL;DR

This paper demonstrates that the Reeb flow on standard contact spheres in higher dimensions has non-trivial elements in the fundamental group of their contactomorphism group, using homotopy properties of contact structures.

Contribution

It establishes the non-triviality of the Reeb flow in the fundamental group of contactomorphisms for high-dimensional spheres, leveraging homotopically non-trivial 2-spheres in contact structure spaces.

Findings

Reeb flow is non-trivial in the fundamental group for spheres with dimension greater than 3.

Homotopically non-trivial 2-spheres exist in the space of contact structures on 3-Sasakian manifolds.

The proof connects contact topology with homotopy theory of contact structure spaces.

Abstract

We prove the non--triviality of the Reeb flow for the (2n+1)--dimensional standard contact spheres inside the fundamental group of their contactomorphism group, n greater than 3. The argument uses the existence of homotopically non--trivial 2--spheres in the space of contact structures of a 3--Sasakian manifold.