Some tight contact foliations can be approximated by overtwisted ones

TL;DR

This paper demonstrates that tightness in contact foliations is not preserved under homotopy, using a turbulisation procedure to approximate tight contact foliations with overtwisted ones.

Contribution

It introduces a turbulisation method showing tightness is not a homotopy invariant in contact foliations, contrasting with known properties in contact topology.

Findings

Tightness is not a homotopy invariant for contact foliations.

A turbulisation procedure can approximate tight foliations by overtwisted ones.

The result impacts understanding of contact structures in foliated manifolds.

Abstract

A contact foliation is a foliation endowed with a leafwise contact structure. In this remark we explain a turbulisation procedure that allows us to prove that tightness is not a homotopy invariant property for contact foliations.