  Starting from Laughlin type wave functions with generalized periodic boundary
conditions describing the degenerate groundstate of a quantum Hall system we
explictly construct $r$ dimensional vector bundles. It turns out that the
filling factor $\nu$ is given by the topological quantity $c_1 \over r$ where
$c_1$ is the first chern number of these vector bundles. In addition, we
managed to proof that under physical natural assumptions the stable vector
bundles correspond to the experimentally dominating series of measured
fractional filling factors $\nu = {n \over 2pn\pm 1}$. Most remarkably, due to
the very special form of the Laughlin wave functions the fluctuations of the
curvature of these vector bundles converge to zero in the limit of infinitely
many particles which shows a new mathematical property. Physically, this means
that in this limit the Hall conductivity is independent of the boundary
conditions which is very important for the observabilty of the effect. Finally
we discuss the relation of this result to a theorem of Donaldson.
