The first Stiefel-Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel Whitney classes for a $d$-dimensional based manifold $M^d$ is defined through $w_1 \in H_1(M^d,\mathbb{Z}_2)$.
The Majorana modes are certain "fermions" whose creation and annihilation operators are the same $\hat{\gamma}=\hat{\gamma}^\dagger$.
For physical reasons, in the context of 2+1 dimensional $p \pm ip$ superconductors, people have a vague impression that Majorana zero modes in the 2+1 dimensional space-time may be related a nontrivial generator in the cohomology group $\mathcal{H}_1(\mathbb{Z}_2,\mathbb{Z}_2)$, where the first $\mathbb{Z}_2$ can be viewed as $\mathbb{Z}_2$ gauge fields, and the second $\mathbb{Z}_2$ has something to do with the orientation of manifold. This may be similar to the context of the first Stiefel-Whitney class, that has something to do with the orientability of the based manifold. In contrast, the second Stiefel-Whitney class $w_2 \in H_2(M^d,\mathbb{Z}_2)$ has something to do with the spin strucutre of the manifold, which is suitable for defining spinors, including the Dirac spinors. Naively, the Dirac (complex) fermion is a pair of Majorana (real) fermions, say in 2+1 dimensional.
A side remark is that $\mathcal{H}_1(\mathbb{Z}_2,\mathbb{Z}_2)$ also occurs in the calculation of the 3rd spin bordism group $\Omega_3^{Spin}(B \mathbb{Z}_2)$ for the classifying space $B \mathbb{Z}_2$.
So, my question is, in mathematics (References/literature are welcome), do the following objects appear in a unified context: (1) Majorana modes/fermions, (2) the first Stiefel Whitney classes $w_1 \in H_1(M^d,\mathbb{Z}_2)$, (3) the cohomology group $\mathcal{H}_1(\mathbb{Z}_2,\mathbb{Z}_2)$? What are the math principles and structures behind?