  We initiate the study of the effects of strongly-coupled gauge interactions
on the properties of the topological phases of matter. In particular, we
discuss fermionic systems with three spatial dimensions, protected by time
reversal symmetry. We first derive a sufficient condition for the introduction
of a dynamical Yang-Mills field to preserve the topological phase of matter,
and then show how the massless pions capture in the infrared the topological
properties of the fermions in the ultraviolet. Finally, we use the S-duality of
${\mathcal N}=2$ supersymmetric $SU(2)$ gauge theory with $N_f{=}4$ flavors to
show that the $\nu{=}16$ phase of Majorana fermions can be continuously
connected to the trivial $\nu{=}0$ phase.
