  We analyze the Landau problem and quantum Hall effect on $S^3$ taking a
constant background field proportional to the spin connection on $S^3$. The
effective strength of the field can be tuned by changing the dimension of the
representation to which the fermions belong. The effective action for the edge
excitations of a quantum Hall droplet in the limit of a large number of
fermions is obtained. We find that the appropriate space for many of these
considerations is $S^2 \times S^2$, which plays a role similar to that of ${\bf
CP}^3$ vis-a-vis $S^4$. We also give a method of representing the algebra of
functions on fuzzy $S^3/{\bf Z}_2$ in terms of finite dimensional matrices.
