On a 3-manifold $Y$, oriented 2-plane fields $\xi$ are oriented rank-2 subbundles of $TY$. Denote the set of such *(up to homotopy)* by $\Theta=\pi\_0\lbrace\xi\rbrace$. **What is an explicit canonical map $\Theta\to\mathbb{Z}\_2$ ?**


In particular, I want to see the canonical mod-2 grading on Seiberg-Witten-Floer Homology, but the classical text of Kronheimer-Mrowka doesn't mention it in terms of 2-plane fields. They construct an isomorphism of $\Theta$ with some abstract set $\mathbb{J}$ based upon "configuration points" $[a]$, and then a canonical map $\mathbb{J}\to\mathbb{Z}\_2$ by assigning to $[a]$ an operator and taking its index mod-2. These two maps are tough enough on their own, so I cannot see what it looks like on 2-plane fields... Plus, there should be some **purely topological partition** of $\Theta$ into two subsets.


Otherwise, perhaps it can be done knowing that oriented 2-plane fields are equivalent to 1-forms of length 1, and are also equivalent to pairs $(\mathfrak{s},\phi)$ where $\mathbb{s}$ is a spin-c structure and $\phi$ is a unit-length spinor. This should then be related to taking a 4-manifold $X$ such that $\partial X=Y$ and with a spin-c structure $\mathfrak{s}\_X$ which extends $\mathfrak{s}$.



$\pi_0${plane fields}$\to\mathbb{Z}_2$
In light of the paper referenced in Tim Perutz' comment, here is the desired map:


An oriented 2-plane field $\xi$ is equivalent to a pair $(\mathfrak{s},\phi)$ on $Y$, where $\mathfrak{s}$ is a spin-c structure and $\phi$ is unit-length spinor. By Proposition 28.1.2 (of Kronheimer-Mrowka's textbook), there exists an oriented 4-manifold $X$ with $\partial X=Y$ and carrying a spin-c structure $\mathfrak{s}\_X=(S^+,\rho\_X)$ which extends $\mathfrak{s}$, i.e. $\mathfrak{s}\cong(S^+|\_Y,\rho\_Y)$. Now the relative Euler class $e(S^+,\phi)$ satisfies $e(S^+,\phi)[X,\partial X]$ = $gr\_z(X,\mathfrak{s}\_X,[a])\in\mathbb{Z}$ for some configuration point $[a]$ associated to $\phi$, and this index is independent of $z$. And by Proposition 28.2.2, this is independent of the choice of $X$ (up to homotopy of $\phi$). This is where the isomorphism $\mathbb{J}(Y)\cong \Theta(Y)$, $\xi\leftrightarrow [a]$, comes from. We can thus write down a map $f:\Theta(Y)\to\mathbb{Z}\_2$ given by
$$\xi\mapsto e(S^+,\phi)[X,\partial X]\;\text{mod}2$$


To be consistent with the literature: When we apply this to $Y=S^3$ we get the flipped even/odd decomposition of $\widehat{HM}(S^3)$. Simply change $f$ by $Aut(\mathbb{Z}\_2)$, i.e. adding a $1$, to get agreement. 


**Addendum:** In fact, this map can be rephrased as the obstruction to finding an almost complex structure $J$ on the above $X$ in such a way that $\xi$ is $J$-invariant.


