Maximal Non-Abelian Gauges and Topology of Gauge Orbit Space

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.Comment: 31 pages, harvmac, 1 figur