  We study the Gribov problem in four-dimensional topological Yang-Mills
theories following the Baulieu-Singer approach in the (anti-)self-dual Landau
gauges. This is a gauge-fixed approach that allows to recover the topological
spectrum, as first constructed by Witten, by means of an equivariant (or
constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories
suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this
has repercussions for this analysis. The resolution of the small
(infinitesimal) gauge copies, in general, affects the dynamics of the
underlying theory. In particular, treating the Gribov problem for the standard
Landau gauge condition in non-topological Yang-Mills theories strongly affects
the dynamics of the theory in the infrared. In the current paper, although the
theory is investigated with the same gauge condition, the effects of the copies
turn out to be completely different. In other words: in both cases, the copies
are there, but the effects are very different. As suggested by the tree-level
exactness of the topological model in this gauge choice, the Gribov copies are
shown to be inoffensive at the quantum level. To be more precise, following
Gribov, we discuss the path integral restriction to the Gribov horizon. The
associated gap equation, which fixes the so-called Gribov parameter, is however
shown to only possess a trivial solution, making the restriction obsolete. We
relate this to the absence of radiative corrections in both gauge and ghost
sectors. We give further evidence by employing the renormalization group which
shows that, for this kind of topological model, the gap equation indeed forbids
the introduction of a massive Gribov parameter.
