I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is defined as follows exactly:

Let $M$ be a closed, oriented, connected 3-manifold, $\xi$ an oriented 2-plane field on $M$, and $(X,J)$ an almost complex manifold such that $\partial X = M $ and $\xi$ is homotopic to the complex tangencies $TM \cap JTM$. If the first Chern class $c_{1}(\xi)$ of $\xi$ is a torsion class, then $\theta (\xi)$ is defined to be $c_{1}^2 (X, J)-2\chi(X)-3\sigma(x)$, where $\chi(X)$ is the Euler characteristic of $X$ and $\sigma(X)$ is the signature of $X$.

In Gompf's paper, he showed that $\theta (\xi)$ is a homotopy invariant of $\xi$ if $c_{1}(\xi)$ is a torsion class.

Here I have two questions.

1. How do we construct such an almost complex manifold $(X, J)$ with a given 2-plane field? Gompf explained a construction of this in Lemma 4.4 of the paper but I cannot understand this construction. He used the Hirzebruch-Hopf obstruction of existence of an almost complex structure on a given "closed" manifold but here we discuss a non-closed manifold.

2. How do we show that if $\theta (\xi_{1}) = \theta (\xi_{2})$ and these 2-plane fields induce the same $spin^c$ structure, $\xi_{1}$ and $\xi_{2}$ are mutually homotopic? Similarly to the former question, Gompf showed this in Proposition 4.16 of the paper, but I cannot understand it. Particularly I want to know the difference $\theta (\xi) - \theta (\xi')$ and how to compute this, where $\xi$ is obtained from $\xi$ by the natural $\mathbb{Z}$-action of the set of 2-plane fields inducing a same $spin^c$ structure.

If you know answers to these questions or, of course, only one question, please tell me that.