  We investigate the quantum Hall problem in the lowest Landau level in two
dimensions, in the presence of an arbitrary number of $\delta$-function
potentials arranged in different geometric configurations. When the number of
delta functions $N_\delta$ is smaller than the number of flux quanta through
the system ($N_\phi$), there is a manifold of $(N_\phi-N_\delta)$ degenerate
states at the original Landau level energy. We prove that the total Chern
number of this set of states is +1 regardless of the number or position of the
$\delta$ functions. Furthermore, we find numerically that, upon the addition of
disorder, this subspace includes a quantum Hall transition which is (in a
well-defined sense) $\textit{quantitatively}$ the same as that for the lowest
Landau level without $\delta$-function impurities, but with a reduced number
$N_\phi' \equiv N_\phi-N_\delta$ of magnetic flux quanta. We discuss the
implications of these results for studies of the integer plateau transitions,
as well as for the many-body problem in the presence of electron-electron
interactions.
