  We present a general formula for the topological part of the effective action
for quantum Hall systems in higher dimensions, including fluctuations of the
gauge field and metric around background fields of a specified topological
class. The result is based on a procedure of integrating up from the Dolbeault
index density which applies for the degeneracies of Landau levels, combined
with some input from the standard descent procedure for anomalies. Features of
the topological action in (2+1), (4+1), (6+1) dimensions, including the
contribution due to gravitational anomalies, are discussed in some detail.
