Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3

TL;DR

This paper proves that certain singular codimension-one foliations on 3-manifolds can be homotoped to regular foliations carried by open books or twisted open books, simplifying Thurston's regularization theorem in dimension 3.

Contribution

It introduces the concept of twisted open books and shows all such singular foliations are homotopic to regular ones carried by these structures, avoiding complex homology arguments.

Findings

Homotopy equivalence to regular foliations via open books

Introduction of twisted open books for singular foliations

Simplified proof of Thurston's regularization theorem in dimension 3

Abstract

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurston's 1976 regularization theorem without using Mather's homology equivalence.