  We study topological properties of classical spherical center vortices with
the low-lying eigenmodes of the Dirac operator in the fundamental and adjoint
representations using both the overlap and asqtad staggered fermion
formulations. In particular we address the puzzle raised in a previous work of
our group [Phys.\ Rev.\ D 77, 14515 (2008)], where we found a violation of the
lattice index theorem with the overlap Dirac operator in the fundamental
representation even for "admissible" gauge fields. Here we confirm the
discrepancy between the topological charge and the index of the Dirac operator
also for asqtad staggered fermions and the adjoint representation. Numerically,
the discrepancy equals the sum of the winding numbers of the spheres when they
are regarded as maps $\mathbbm R^3 \cup \{\infty\} \to SU(2)$. Furthermore we
find some evidence for fractional topological charge during cooling the
spherical center vortex on a $40^3 \times 2$ lattice. The object with
topological charge $Q=1/2$ we identify as a Dirac monopole with a gauge field
fading away at large distances. Therefore even for periodic boundary conditions
it does not need an antimonopole.
