The generalized Harer conjecture for the homology triviality

The classical Harer conjecture is about the stable homology triviality of the obvious embedding $\phi :B_{2g+2}\hookrightarrow {\mathrm{\Gamma }}_g$, which was proved by Song and Tillmann. The main part of the proof is to show that $\text{B}{\phi }^{+}:\text{B}{B}_{\mathrm{\infty }}^{+}\to \text{B}{\mathrm{\Gamma }}_{\mathrm{\infty }}^{+}$ 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 $\phi :B_n\hookrightarrow {\mathrm{\Gamma }}_{g,k}$. 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 $\mathrm{\Phi }:\mathcal{C}\to \mathcal{S}$ induced by $\text{B}\phi :{\text{conf}}_n(D)\to {\mathcal{M}}_{g,k}$ preserves the actions of the framed little 2-disks operad.