A moment of cohomology$.$ As is well-known (cf. Ref.1), the momentum operator is defined up to a time-independent closed form. More precisely, the physically inequivalent momentum operators are classified by the de Rham cohomology classes of the configuration space which, if topologically non-trivial, contains more that one element ($b_1$ elements, the first Betti number).
But an actual physical system has a unique momentum operator. Different elements of $H^1$ correspond to different dynamics; but the dynamics of an actual system are fixed – they are what they are. Therefore, what determines which momentum operator is actually realised in a particular system? If there is no principle that allows us to determine, a priori, which momentum operator is "the correct one", can we at least do so a posteriori? Has such an experiment ever been performed?
References.


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*DeWitt - The global approach to quantum field theory, chapter 11.

 A: De-Witt defines the momentum operator as:
$$p_i = i \frac{\partial}{\partial x^i}-\omega_i(x,t)$$
He requires the form $\omega = \omega_i dx^i$ to be closed in order not to alter momentum-momentum canonical commutation relations.
Please notice that the momentum operator describes a minimal coupling to an Abelian vector potential (a U(1) connection) with vanishing field strength. This is a pure gauge electromagnetic vector potential or in the mathematical terminology a flat connection. 
The pure gauge vector potential has a nontrivial effect on the dynamics (i.e. the solutions of the Schrodinger equation) only if the manifold on which the particle lives has a nontrivial fundamental group $\pi_1(M)$. Please, see the following lecture notes by V.P. Nair section 4(a) where the same phenomenon is explained from a somewhat more modern point of view.
Of course, (when there is a nontrivial solution for the vector potential) the system is an Aharonov-Bohm type of state. This means that there is a source (solenoid) generating this vector potential. The nontriviality of the momentum operator can thus be experimentally detected by an Aharonov-Bohm interference experiment.
The inequivalent quantizations of this system are parametrized by the flux of the solenoid modulo the flux causing an Aharonov-Bohm phase of $2 \pi$. That is the space of inequivalent quantizations is $\mathbb{R}/ 2\pi\mathbb{Z}$. A single system (a particle moving in the solenoid field) has a single momentum operator corresponding to a single point in the space of inequivalent quantizations. In order to realize another momentum operator, one can, for example, add another particle moving around its own solenoid. The second particle will have a momentum operator with a different vector potential. Another possibility is to change the flux in the solenoid, as a result another possibility of the momentum operator will be realized. The latter possibility is a topology change of the flat principal Aharonov-Bohm bundle  .
Remarks: 


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*This effect detects co(homology) rather than homotopy,  when the particle is scalar. The first homology group is the Abelianization of the fundamental group;  in order to detect the full fundamental group (i.e. find a connection with nontrivial holonomy on all the generators of the group), we need a particle with internal degrees of freedom. This is related to the 
recent question on PSE that you have participated in. The latter is sometimes called, the non-Abelian Aharonov-Bohm effect.

*Non-trivial (Abelian, non-Abelian, generalized) pure gauge fields (and their spaces of solutions) describe important physical effects. For example, they represent gauge equivalence classes of solutions of the Chern-Simons model.

*This type of topological effect was treated in a few occasions, from different point of views, on PSE, please see for example, these two questions that I have answered.
