  We provide a characterization of tunneling between coupled topological
insulators in 2D and 3D under the influence of a ferromagnetic layer. We
explore conditions for such systems to exhibit integer quantum Hall physics and
localized fractional charge, also taking into account interaction effects for
the 2D case. We show that the effects of tunneling are topologically equivalent
to a certain deformation or folding of the sample geometry. Our key advance is
the realization that the quantum Hall or fractional charge physics can appear
in the presence of only a \emph{single} magnet unlike previous proposals which
involve magnetic domain walls on the surface or edges of topological insulators
respectively. We give illustrative topological folding arguments to prove our
results and show that for the 2D case our results are robust even in the
presence of interactions.
