  The fundamental group and rational cohomology of the configuration spaces of
the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be
a compact oriented 3-manifold, either with or without a marked point
representing an end at infinity. For the Skyrme model, the codomain is any Lie
group, while for the Faddeev-Hopf model it is $S^2$. It is determined when the
topology of configuration space permits fermionic and isospinorial quantization
of the solitons of the model within generalizations of the frameworks of
Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is
possible only if the target group contains a symplectic or special unitary
factor, while fermionic quantization of Hopfions is always possible. Geometric
interpretations of the results are given.
