Discrete Pin structures

Question

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.

1. P. W. Kasteleyn, Dimer Statistics and Phase Transitions Journal of Mathematical Physics 4, 287 (1963); https://doi.org/10.1063/1.1703953

2. Richard Z Goldstein and Edward C Turner, “A formula for Stiefel-Whitney homology classes,” Proceedings of the American Mathematical Society 58, 339–339 (1976).

see also:

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

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

Answer 1

Here are some partial answers.

For your first question: there are combinatorial formulas for all Stiefel-Whitney homology classes $w_k$, due to Whitney and rediscovered by Cheeger. Specifically, on a manifold with a triangulation $\Pi$, $w_k$ is represented by the sum of all $k$-simplices in the barycentric subdivision of $\Pi$.

So the story you have for spin 2-manifolds and $w_0$ should generalize for a pin^$+$ $n$-manifold, where you want a trivialization of $[w_{n-2}]$, and can represent it using Whitney's combinatorial formula. For a pin^$-$ $n$-manifold, though, you'd need a trivialization of the Poincaré dual of $(w^1)^2$, which is not calculated by Whitney's formula. It may be possible to imitate Whitney's argument for $(w^1)^2$, but I don't think it's trivial to do so.

For your second question: in “Dimers on graphs in non-orientable surfaces”, §§4–5, David Cimasoni describes a generalization of Kastelyn orientations to pin^$-$ surfaces, and proves in Theorem 5.3 that given a cell decomposition and a dimer configuration on a closed surface $\Sigma$, generalized Kastelyn orientations are equivalent to pin^$-$-structures on $\Sigma$.

As far as I know, however, nothing has been written generalizing this past dimension 2, nor about combinatorial pin^$+$-structures in any dimension.

This question is related to another MathOverflow question from a few years ago.