  Under general assumptions, we present a low-energy effective action for the
quantum Hall state when edges exist. It is shown that the chiral edge current
is necessary to make the effective action to be gauge invariant. However the
chiral edge current is irrelevant to the Hall current. The exactly quantized
value of $\sigma_{xy}$ is observed only when the Hall current does not flow at
the edge region. Our effective theory is applicable to the quantum Hall liquid
on a surface with non-trivial topology and physical meanings of the topology
are discussed.
