  When we impose the discrete symmetry in the standard model we have Dai-Freed
global anomalies. However, interestingly if we introduce three right-handed
neutrinos we can have an anomaly-free discrete $Z_4$ gauge symmetry. This $Z_4$
symmetry should be spontaneously broken down to the $Z_2$ symmetry to generate
the heavy Majorana masses for the right-handed neutrinos. We show that this
symmetry breaking naturally generates topological inflation, which is
consistent with the CMB observations at present and predicts a significant
tensor mode with scalar-tensor ratio $r > 0.03$. The right-handed neutrinos
play an important role in reheating processes. The reheating temperature is as
high as $\sim 10^8$GeV, and non-thermal leptogenesis successfully takes place.
