Aharonov–Bohm effect in brief is due to some singularities in space. In books it's infinite solenoid most of the time, which makes some regions of space not simply connected.
What intrigues me is the fact that in real experiment we can't use infinite solenoid. So even if we use one and say that locally it's good approximation it doesn't change the fact that whole space is still simply connected. But the fact is that the effect was experimentally observed.
So the question arises - how one should describe this effect in more rigorous manner (or maybe not rigorous but possible in real world)?