It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of the simplicial complex upon triangulation of $M$, one can also use the $(n-2)$-th Stiefel-Whitney homology class $[w_{n-2}]$, which is the Poincare dual of $[w^2]$.

There is a relation between the discrete spin structure and Kasteleyn orientations, given in Kasteleyn (Ref 1).

For a spatial 2-manifold $M^2$ ($n=2$) with triangulation $T$, the Stiefel-Whitney homology class $[w_0]$ has a representative that is the summation of all vertices $v$ with some (mod 2) coefficients as follows Goldstein- Turner (Ref 2):

$$ w_0 = \sum_{v\in T} \# \{\sigma | v \subseteq \sigma \text{ is regular} \} \cdot v. $$

Here, $v \subseteq \sigma$ means that $v$ is a sub-simplex of simplex $\sigma$. The subsimplex $v \subseteq \sigma$ is called regular if $v$ and $\sigma$ satisfy the certain relative positions. The $\# \{\sigma | v \subseteq \sigma \text{ is regular} \} \cdot v$ denotes the formal product of the (mod 2) number of regular pairs $v \subseteq \sigma$ and the vertex $v$.

One can call vertex $v$ singular if $\# \{\sigma | v \subseteq \sigma \text{ is regular} \}$ is odd. Then $w_0$ is the formal summation of all singular vertices. $w_0$ is a vector (0-th singular chain) in the vector space (of 0-th singular chains) spanned by the formal bases of all vertices with $\mathbb Z_2$ coefficients.

Note any 2D oriented manifold allow spin structures.

Question:

(1) Are there analogous discrete Pin structures (say Pin$^+$ or Pin$^-$) that we can define as "Stiefel-Whitney homology classes" for non-orientable $n$-manifolds $M$. Say $n=3, 4, 5$?

where Pin structure is given by: $$ 1\to \mathbb{Z}_2 \to \text{Pin}^{\pm}(n) \to \text{Spin}(n) \to 1 $$

In terms of cohomology class, for Pin$^+$, $w^2(M)=0$; and for Pin$^-$, $w^2(M)+(w^1)^2(M)=0$. (Here the usual notation shall be, for Pin$^+$, $w_2(M)=0$; and for Pin$^-$, $w_2(M)+w_1^2(M)=0$. )

(2) What is the counterpart of Kasteleyn orientations, in $$\text{Kasteleyn orientations v.s. discrete Spin structures}$$ $$\simeq \text{??? v.s. discrete Pin structures?}$$

References are welcome.

see also:

David Cimasoni and Nicolai Reshetikhin, “Dimers on Surface Graphs and Spin Structures. i,” Communications in Mathematical Physics 275, 187–208 (2007).

David Cimasoni and Nicolai Reshetikhin, “Dimers on Surface Graphs and Spin Structures. II,” Communications in Mathematical Physics 281, 445–468 (2008).