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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.

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    $\begingroup$ Have you looked at Kronheimer and Mrowka's book Monopoles and 3-Manifolds? $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 14, 2014 at 21:48
  • $\begingroup$ There's also a nice discussion in Kronheimer-Mrowka's paper "Monopoles and contact structures." $\endgroup$
    – Tim Perutz
    Commented Nov 15, 2014 at 0:16
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    $\begingroup$ I have tried to read Kronheimer and Mrowka's book "Monopoles and 3-Manifolds." An answer of my questions is Proposition 28.1.2 in this book, isn't it? Although I have understood substantially a construction of an almost complex manifold we want, I've not understood how to extand this structure to the 4-manifold with boundary $S^1 \times S^2$. Could you explain that to me? $\endgroup$
    – Math1016
    Commented Nov 15, 2014 at 6:49
  • $\begingroup$ Look at page 194 of this book: "Surgery on contact 3-manifolds and Stein structures" by Ozbagci and Stipsicz. They explain how to build a 4-manifold $(X,J)$ with $\partial X=M$. This is the rough idea: start with a contact surgery link of $M$. Look at the corresponding 4-manifold $X$ obtained by attaching 2-handles to this link. $X$ should admit an achiral Lefschetz fibration. Away from the singular points, let $J$ be $\pi/2$-rotation on the tangent plane of fibers. Now we can extend $J$ to neighborhoods of the singular points, since there are honest complex charts near them. $\endgroup$
    – nikita
    Commented Nov 24, 2014 at 23:13
  • $\begingroup$ There is a discussion in the same book about two contact structures are homotopic iff their so-called $d_2$ and $d_3$ (the last one is the same as $\theta$ in your notation) invariants are equal. $\endgroup$
    – nikita
    Commented Nov 24, 2014 at 23:27

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