  A function from configuration space to moduli space of surface may induce a
homomorphism between their fundamental groups which are braid groups and
mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow
\Gamma_{g,b}$ is induced by 3-fold branched covering over a disk with some
branch points. In this thesis we give a concrete description of this map and
show that it is injective by Birman-Hilden theory. This gives us a new
interesting non-geometric embedding of braid group into mapping class group. On
the other hand, we show that the map on the level of classifying spaces of
groups is compatible with the action of little 2-cube operad so that it induces
a trivial homomorphim between stable homology group of braid groups and that of
mapping class groups(Harer conjecture). We also show how the lift
$\tilde{\beta_i}$ acts on the fundamental group of the surface and through this
we prove that $\tilde{\beta_i}$ equals the product of two inverse Dehn twists.
