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Register a new userHaefligers codimension-one singular foliations, open books and twisted open books in dimension 3
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Authors
Francois Laudenbach
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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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We prove that if a contact manifold $(M,xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.
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