  We introduce two maximal non-abelian gauge fixing conditions on the space of
gauge orbits M for gauge theories over spaces with dimensions d < 3. The gauge
fixings are complete in the sense that describe an open dense set M_0 of the
space of gauge orbits M and select one and only one gauge field per gauge orbit
in M_0. There are not Gribov copies or ambiguities in these gauges. M_0 is a
contractible manifold with trivial topology. The set of gauge orbits which are
not described by the gauge conditions M \ M_0 is the boundary of M_0 and
encodes all non-trivial topological properties of the space of gauge orbits.
The gauge fields configurations of this boundary M \ M_0 can be explicitly
identified with non-abelian monopoles and they are shown to play a very
relevant role in the non-perturbative behaviour of gauge theories in one, two
and three space dimensions. It is conjectured that their role is also crucial
for quark confinement in 3+1 dimensional gauge theories.
