  The quantum Hall effect is studied in a spherical geometry using the Dirac
operator for non-interacting fermions in a background magnetic field, which is
supplied by a Wu-Yang magnetic monopole at the centre of the sphere. Wave
functions are cross-section of a non-trivial $U(1)$ bundle, the zero point
energy then vanishes and no perturbations can lower the energy. The
Atiyah-Singer index theorem constrains the degeneracy of the ground state.
  The fractional quantum Hall effect is also studied in the composite Fermion
model. Vortices of the statistical gauge field are supplied by Dirac strings
associated with the monopole field. A unique ground state is attained only if
the vortices have an even number of flux units and act to counteract the
background field, reducing the effective field seen by the composite fermions.
There is a unique gapped ground state and, for large particle numbers,
fractions $\nu=\frac{1}{2 k+1}$ are recovered.
