In trying to get the Aharonov-effect right I've found something that I'm not sure how to sort out. Briefly put my understanding of the effect is that it shows something that cannot be explained by classical physics in the sense that makes observable a classical EM global gauge transformation that shouldn't be observable within classical EM, where only the effects of the fields are observable but not the effects of the potentials. But this is not the case in QM where potentials are physical, and so it was theoretically predicted and later experimentally confirmed that charged particles going thru a a region where a magnetic field is negligible show a shift in their diffraction pattern caused just by the vector potential and varying with the flux thru the solenoid. Now there is no question about where the shift comes from because the pattern is different between a close to zero magnetic field and a potential. So in practice there is no need to really make the magnetic field vanishing with an ideal infinite solenoid, which is great because otherwise the experiment would be impossible. The explanation of the effect as given in topological terms is that the presence of the solenoid inside the loop-like disposition of the electrons makes the topology of the space nontrivial, and this is what causes the potential to be observable. Specifically the presence of a string defect makes the winding number around it observable. This is because R^3 space minus an infinite line makes the space no longer simply-connected. In practice the experimental setup obviously doesn't use an infinite solenoid, since it is considered that the space in wich the electrons are confined, kind of a torus-like, is also not simply-connected and that is what matters. Now to the part I find difficult, it is my understanding that the topological requirement for the existence of a vector potential in the presence of a magnetic field when the gaus law for magnetism holds is that the space must be contractible,and contractibility implies simply-connectedness, so if the experimental space set up for the A-B effect to show up must be non-simply connected, it would seem like the very condition for the existence of the vector potential is absent. Any hints?