  Using the Dirac and the Yang monopole in spinor condensates as examples, we
show that interactions can stretch the point singularity of a monopole into an
extended manifold, whose shape is strongly influenced by the sign of
interaction. The singular manifold will cause the first and second Chern number
to assume non-integer values when it intersects the surface on which the Chern
numbers are calculated. This leads to a gradual decrease of the Chern numbers
as the monopole moves away from the surface of integration, instead of the
sudden jump characteristic of a point monopole. A gradual change in $C_2$ has
in fact been observed in the recent experiment by Spielman's group at NIST. By
measuring the range of non-integer values of the Chern numbers as the monopole
moves away from the surface of integration along different directions, one can
map out the shape of the singular manifold in the parameter space.
