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A function from configuration space to moduli space of surface may induce ahomomorphism between their fundamental groups which are braid groups andmapping 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 somebranch points. In this thesis we give a concrete description of this map andshow that it is injective by Birman-Hilden theory. This gives us a newinteresting non-geometric embedding of braid group into mapping class group. Onthe other hand, we show that the map on the level of classifying spaces ofgroups is compatible with the action of little 2-cube operad so that it inducesa trivial homomorphim between stable homology group of braid groups and that ofmapping class groups(Harer conjecture). We also show how the lift$\tilde{\beta_i}$ acts on the fundamental group of the surface and through thiswe prove that $\tilde{\beta_i}$ equals the product of two inverse Dehn twists.

Configuration space, moduli space and 3-fold covering space