  For an $S_4$ space-time manifold global aspects of gauge-fixing are
investigated using the relation to Topological Quantum Field Theory on the
gauge group. The partition function of this TQFT is shown to compute the
regularized Euler character of a suitably defined space of gauge
transformations. Topological properties of the space of solutions to a
covariant gauge conditon on the orbit of a particular instanton are found using
the $SO(5)$ isometry group of the $S_4$ base manifold. We obtain that the Euler
character of this space differs from that of an orbit in the topologically
trivial sector. This result implies that an orbit with Pontryagin number
$\k=\pm1$ in covariant gauges on $S_4$ contributes to physical correlation
functions with a different multiplicity factor due to the Gribov copies, than
an orbit in the trivial $\k=0$ sector. Similar topological arguments show that
there is no contribution from the topologically trivial sector to physical
correlation functions in gauges defined by a nondegenerate background
connection. We discuss possible physical implications of the global gauge
dependence of Yang-Mills theory.
