  We show that, in a model of modified gravity based on the spectral action
functional, there is a nontrivial coupling between cosmic topology and
inflation, in the sense that the shape of the possible slow-roll inflation
potentials obtained in the model from the nonperturbative form of the spectral
action are sensitive not only to the geometry (flat or positively curved) of
the universe, but also to the different possible non-simply connected
topologies. We show this by explicitly computing the nonperturbative spectral
action for some candidate flat cosmic topologies given by Bieberbach manifolds
and showing that the resulting inflation potential differs from that of the
flat torus by a multiplicative factor, similarly to what happens in the case of
the spectral action of the spherical forms in relation to the case of the
3-sphere. We then show that, while the slow-roll parameters differ between the
spherical and flat manifolds but do not distinguish different topologies within
each class, the power spectra detect the different scalings of the slow-roll
potential and therefore distinguish between the various topologies, both in the
spherical and in the flat case.
