Relating the topological behaviour in the toric code to cohomology?

I’ve been working on the Toric Code Model (by Kitaev in his 2003 paper on quantum computation), and the model is a lattice realisation of a topological phase.

The local operators in the model are given by: $sigma^x_s, sigma^z_s, A_v = Pi_{s in +} sigma^x_s, B_p = Pi_{s in Box} sigma^z_s $. Where s is a lattice edge, v is a vertex, p is a plaquette.

Now the ground state of this model is given by $|Omega rangle$ such that $A_v |Omega rangle = B_p |Omegarangle = | Omega rangle$, and the excited state consisting of a single electrical excitation is given by $F_gamma |Omega rangle$ where $F_gamma = Pi_{sin gamma}
sigma^z_s$ and the path $gamma$ is a semi infinite path stretching unto infinity. We can for the sake of concreteness say that the path is stretching infinitely in the positive x direction.

With some of the prescription out of the way, I wanted to enquire about the relation of the topological nature of this model with the cohomological description of topology that I’m used to.

If we deform the path by introducing a kink to it, all the local observables on the electric excitation will not be able to detect this kink, provided it isn’t at the end of the path.

This is why it is a topological model: deforming this path slightly does not change anything about the system (all the local observables are blind to this change).

However, I was wondering if somehow it is related to cohomology? Like maybe showing that if for every $R_{gamma’} = R_{gamma} + R_{kink}$, where R is a semi infinite path on the lattice, but $R_{kink} = d (text{something else})$ and thus the kink is an element that is closed and exact in the cohomology of something? Forgive me if I’m not making sense with this notation, I’m not too familiar with this math. Maybe if I could possibly get an in depth explanation of how the two ways of talking about topology meld together it would be nice!