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

Abstract

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by Andre Haefliger as being Γ1-structures. We prove that under the obvious linear embedding condition, they are Γ1-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 Cr, r ≥ 1. In particular, in dimension 3, this gives a very simple proof of Thurston’s 1976 regularization theorem without using Mather’s homology equivalence.