  We investigate the topology of the space of M\"obius conjugacy classes of
degree $d$ rational maps on the Riemann sphere. We show that it is rationally
acyclic and we compute its fundamental group. As a byproduct, we also obtain
the ranks of some higher homotopy groups of the parameter space of degree $d$
rational maps allowing us to extend the previously known range. Moreover, we
show that this parameter space is not nilpotent.
