  We use the two-dimensional Schwinger model to investigate how lattice
fermions perceive the global topological charge $q\in\mathbf{Z}$ of a given
gauge background $U$. After a warm-up part devoted to staggered, Adams, Wilson
and naive fermions, we consider Karsten-Wilczek and Borici-Creutz fermions,
which are in the class of minimally doubled lattice fermion actions. We focus
on the eigenvalue spectrum and the chiralities of the pertinent eigenmodes.
Without modification both minimally doubled actions are found to be insensitive
to topology, but in either case it is possible to define a suitable
species-splitting term to make the resulting operator topology aware.
