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Regularization of Gamma_1-structures in dimension 3

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 Added by Francois Laudenbach
 Publication date 2009
  fields
and research's language is English




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For $Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurstons regularization theorem, first proved in cite{thurston}, without using Mathers homology equivalence. Moreover, in the co-orientable case, the resulting foliation can be chosen of a precise kind, namely an open book foliation modified by suspension. There is also a model in the non co-orientable case.



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We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every homotopy class of Haefliger structures, in every differentiability class, under the obvious embedding assumption. The proof uses only elementary means, and reproves Thurstons existence theorem in all dimensions. A parametric version is also established.
73 - Michael Harrison 2019
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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 Thurstons 1976 regularization theorem without using Mathers homology equivalence.
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