I came across the following statement when reading Section 4 of this paper:
For an oriented 2-plane field $\xi$, let $d(\xi) \in \mathbb{Z}$ denote the divisibility of the Chern class, so that $c_1(\xi)$ equals $d(\xi)$ times a primitive class in $H^2(M;\mathbb{Z})$ modulo torsion, and $d(\xi) = 0$ if $c_1(\xi)$ is of finite order.
The grammar and clarity can be improved as follows:
Here, $M$ is a closed, connected 3-manifold. While viewing $H^2(M;\mathbb{Z})$ as an abelian group, I'm uncertain about the terms "primitive class" and "divisibility mod torsion." Could someone clarify these notions? Thank you.
I don't think there is anything to improve here. It is very clear what $d(\xi)$ is.
Consider a torsion-free abelian group $G$, and a nonzero element $g\in G$:
the divisibility $d(g)$ is the largest integer $n\geq 1$ such that there exists $h\in G$ with $g=nh$. (If for every $n$ you can find $h$, we say $g$ is infinitely divisible.)
$g$ is primitive if $d(g)=1$, i.e., it is not divisible by any integer $>1$.
Now you have the (finitely-generated) abelian group $H^2(M;\mathbb{Z})$, so you can mod out its torsion subgroup to get yourself a torsion-free abelian group and consider the image of $c_1(\xi)$ in $H^2(M;\mathbb{Z})/H^2(M;\mathbb{Z})_{\mathrm{tors}}\cong\mathbb{Z}^{b^2(M)}$ and get $d(\xi):=d(c_1(\xi)+H^2(M;\mathbb{Z})_{\mathrm{tors}})$. Finally we also deal with the case $c_1(\xi)\in H^2(M;\mathbb{Z})_{\mathrm{tors}}$.