I am a beginner in Contact Geometry.

To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes.

This is what I could do so far:

Start with an orientable 2-plane field $\xi$ on a 3-manifold. Triangulate $M$ such that on each 3-simplex $\xi$ is transverse to the edges as well as the faces in a manner that on each 3-simplex, $\xi$ is $C^\infty$-close to the foliation $dz=0$. If there is a tangency point $p$ on the interior of some edge $e$, then we can assume it to be generic, and then one subdivides the 3-simplex by a plane that passes through $p$ and the edge that does not intersect $e$.

Question 1: Why a generic tangency is needed at $p$? I somehow can't see why merely a tangency would do.

Next homotop $\xi$ to a contact structure successively over 0-, 1- and 2- skeletons.

Then we're left with 3-balls inside the interior of each 3-simplex. We use a trick to turn the problem into extending over an S^1\times D^2 by taking a curve outside the 3-ball transverse to $\xi$, but with end points on the boundary of the 3-ball, and thickening the curve so we get a copy of $S^1\times D^2$ (open solid torus)

In this torus $\xi$ is isotopic to a foliation by planes $pt\times D^2$. So we can then perform the Lutz twist on the foliation $\cos 2\pi r dz + r\sin 2\pi r d\theta$.

Question 2: How to ensure that $\xi$ is contact on $\partial(S^1\times D^2)$ after performing the twist? I tried to get $\eta$ which is contact on a neighbourhood of $\partial(S^1\times D^2)$ agrees with $\xi$ outside it, but in vain.


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