  We obtain high-velocity estimates with error bounds for the scattering
operator of the Schr\"odinger equation in three dimensions with electromagnetic
potentials in the exterior of bounded obstacles that are handlebodies. A
particular case is a finite number of tori. We prove our results with
time-dependent methods. We consider high-velocity estimates where the direction
of the velocity of the incoming electrons is kept fixed as its absolute value
goes to infinity. In the case of one torus our results give a rigorous proof
that quantum mechanics predicts the interference patterns observed in the
fundamental experiments of Tonomura et al. that gave a conclusive evidence of
the existence of the Aharonov-Bohm effect using a toroidal magnet. We give a
method for the reconstruction of the flux of the magnetic field over a
cross-section of the torus modulo $2\pi$. Equivalently, we determine modulo
$2\pi$ the difference in phase for two electrons that travel to infinity, when
one goes inside the hole and the other outside it. For this purpose we only
need the high-velocity limit of the scattering operator for one direction of
the velocity of the incoming electrons. When there are several tori -or more
generally handlebodies- the information that we obtain in the fluxes, and on
the difference of phases, depends on the relative position of the tori and on
the direction of the velocities when we take the high-velocity limit of the
incoming electrons. For some locations of the tori we can determine all the
fluxes modulo 2$\pi$ by taking the high-velocity limit in only one direction.
We also give a method for the unique reconstruction of the electric potential
and the magnetic field outside the handlebodies from the high-velocity limit of
the scattering operator.
