  We report the discovery of flatten Bloch bands with nontrivial topological
numbers in a quasi-one-dimensional sawtooth chain. We present the nearly
flat-band with a obvious gap and nonzero Chern number of the modulated sawtooth
chain by tuning hoppings and modulation. With the increasing of the strength of
the modulation, the system undergos a series of topological phase transitions
in the absence of interaction. By adding interaction to the model, we present
exact diagonalization results for the system at $1/3$ filling and some
interesting connections with fraction quantum Hall states: topological
degeneracy, nontrivially topological number and fractional statistic of the
quasihole.
