  The classical Harer conjecture is about the stable homology triviality of the
obvious embedding $\phi : B_{2g+2} \hookrightarrow \Gamma_{g}$, which was
proved by Song and Tillmann. The main part of the proof is to show that
$\B\phi^{+} : \B B_{\infty}^{+} \rightarrow \B \Gamma_{\infty}^{+}$ induced
from $\phi$ is a double loop space map. In this paper, we give a proof of the
generalized Harer conjecture which is about the homology triviality for an
$arbitrary$ embedding $\phi : B_{n} \hookrightarrow \Gamma_{g,k}$. We first
show that it suffices to prove it for a $regular$ embedding in which all atomic
surfaces are regarded as identical and each atomic twist is a {\it simple
twist} interchanging two identical sub-parts of atomic surfaces. The main
strategy of the proof is to show that the map $\Phi : \mathcal{C} \rightarrow
\mathcal{S}$ induced by $\B\phi:\conf_n(D)\rightarrow\mathcal{M}_{g,k}$
preserves the actions of the framed little 2-disks operad.
