Maximal Non - Abelian Gauges and Topologyof Gauge Orbit

We introduce two maximal non-abelian gauge xing conditions on the space of gauge orbits M for gauge theories over spaces with dimensions d 3. The gauge xings 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 eld 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 n M 0 is the boundary of M 0 and encodes all non-trivial topological properties of the space of gauge orbits. The gauge elds conngurations of this boundary M n M 0 can be explicitly identiied 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 connnement in 3+1 dimensional gauge theories.