  Given a smooth, oriented, closed surface $\Sigma$ of genus zero, possibly
with boundary, let $\tilde{\Sigma} \longrightarrow \Sigma$ be a given $G$-cover
of $\Sigma$, where $G$ is a given finite group. Let $S_{n}$ denote the standard
sphere with $n$ holes. There are many ways of gluing together several $G$-cover
of $S_{n}$ to construct the $G$-cover $\ts \longrightarrow \Sigma$, of
$\Sigma$. We let $M(\tilde{\Sigma} ,\Sigma)$ be the set of all ways to
construct the given $G$-cover, $\tilde{\Sigma} \longrightarrow \Sigma$, of
$\Sigma$ from gluing of several $G$-covers of $S_{n}$, here $n$ may vary. In
this paper, we define some simple moves and relation which will turn
$M(\tilde{\Sigma} ,\Sigma)$ into a connected and simply-connected complex. This
will be used in the future paper to construct $G$-equivariant Modular Functor.
This $G$-equivariant Modular Functor will be an extension of the usual Modular
Functor.
