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 $\mathfrak{s}$.

the existenceof such a map? I could see trying to construct a map from this, by taking $X$ such that $\partial X=Y$ and then removing a ball from it to get a cobordism $W:Y\to S^3$, giving a parity-relation between $\Theta(Y)$ and $\Theta(S^3)$ via that ``cobordism index'' $\iota(W)$, but then we had to a priori decide on $\Theta(S^3)\to\mathbb{Z}_2$. $\endgroup$ – Chris Gerig Apr 20 '12 at 6:23