I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following statement (Lemma

"Let $\xi$ be a simple contact structure near the boundary of $S = \partial B$ of the 3-ball. Then the extendability of $\xi$ as a contact structure to $B$ depends only on the topological type of the foliation on $S$ induced by $\xi$."

I understand that a contact structure defined in the neighborhood of a surface is basically defined by the diffeomorphism class of the foliation it induces. (This is a theorem of Giroux?) Hence the main point is to pass from "topological type" (i.e., homeomorphism class) to diffeomorphism class. To do this, Eliashberg gives some sort of perturbation argument. However, I'm not entirely sure I understand what he has written.

First of all, he starts off by choosing what seems to be a transversal to the foliation, but I am not sure this is possible, (Transversals exist for almost horizontal foliations, but maybe not simple ones?) He then makes what I think are several typographical errors. I would be very thankful if someone could explain the details of the proof to me - it isn't very long, but has left me quite confused.

Finally, what is an example of two simple foliations on $S^2$ that are homeomorphic but not diffeomorphic (i.e., why is the lemma necessary)? Sorry; I'm not terribly familiar with foliation theory.

Thanks in advance!


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