  Systems of free fermions are classified by symmetry, space dimensionality,
and topological properties described by K-homology. Those systems belonging to
different classes are inequivalent. In contrast, we show that by taking a
many-body/Fock space viewpoint it becomes possible to establish equivalences of
topological insulators and superconductors in terms of duality transformations.
These mappings connect topologically inequivalent systems of fermions, jumping
across entries in existent classification tables, because of the phenomenon of
symmetry transmutation by which a symmetry and its dual partner have identical
algebraic properties but very different physical interpretations. To constrain
our study to established classification tables, we define and characterize
mathematically Gaussian dualities as dualities mapping free fermions to free
fermions (and interacting to interacting). By introducing a large, flexible
class of Gaussian dualities we show that any insulator is dual to a
superconductor, and that fermionic edge modes are dual to Majorana edge modes,
that is, the Gaussian dualities of this paper preserve the bulk-boundary
correspondence. Transmutation of relevant symmetries, particle number,
translation, and time reversal is also investigated in detail. As illustrative
examples, we show the duality equivalence of the dimerized Peierls chain and
the Majorana chain of Kitaev, and a two-dimensional Kekul\'e-type topological
insulator, including graphene as a special instance in coupling space, dual to
a p-wave superconductor. Since our analysis extends to interacting fermion
systems we also briefly discuss some such applications.
