  Mappings between models may be obtained by unitary transformations with
preservation of the spectra but in general a change in the states. Non-
canonical transformations in general also change the statistics of the
operators involved. In these cases one may expect a change of topological
properties as a consequence of the mapping. Here we consider some dualities
resulting from mappings, by systematically using a Majorana fermion
representation of spin and fermionic problems. We focus on the change of
topological invariants that results from unitary transformations taking as
examples the mapping between a spin system and a topological superconductor,
and between different fermionic systems.
