  In this paper we consider two generalizations of the Skyrme model. One is a
variational problem for maps from a compact three-manifold to a compact Lie
group. The other is a variational problem for flat connections. We describe the
path components of the configuration spaces of smooth fields for each of the
variational problems. We prove that the invariants separating the path
components are well-defined for (not necessarily smooth) fields with finite
Skyrme energy. We prove that for every possible value of these invariants there
exists a minimizer of the Skyrme functional. Throughout the paper we emphasize
the importance of holonomy in the Skyrme model. Some of the results may be
useful in other contexts. In particular, we define the holonomy of a
distributionally flat $ L^2_{loc} $ connection; the local developing maps for
such connections need not be continuous.
