  In this paper we consider a generalization of the Faddeev model for the maps
from a closed three-manifold into the two-sphere. We give a novel
representation of smooth $ S^2$-valued maps based on flat connections. This
representation allows us to obtain an analytic description of the homotopy
classes of $ S^2$-valued maps that generalizes to Sobolev maps. It also leads
to a new proof an old theorem of Pontrjagin. For the generalized Faddeev model,
we prove the existence of minimizers in every homotopy class.
