  Subtle issues arise when extending homotopy invariants to spaces of functions
having little regularity, e.g., Sobolev spaces containing discontinuous
functions. Sometimes it is not possible to extend the invariant at all, and
sometimes, even when the formulas defining the invariants make sense, they may
not have expected properties (e.g., there are maps having non-integral degree).
  In this paper we define a complete set of homotopy invariants for maps from
3-manifolds to the 2-sphere and show that these invariants extend to finite
Faddeev energy maps and maps in suitable Sobolev spaces. For smooth maps, our
description is proved to be equivalent to Pontrjagin's original homotopy
classification from the 1930's. We further show that for the finite energy maps
the invariants take on exactly the same values as for smooth maps. We include
applications to the Faddeev model.
  The techniques that we use would also apply to many more problems and/or
other functionals. We have tried to make the paper accessible to analysts,
geometers and mathematical physicists.
