The generalized Harer conjecture for the homology triviality

Abstract

The classical Harer conjecture is about the stable homology triviality of the obvious embedding $ϕ: B_{2g+2} \hookrightarrow Γ_{g}$, which was proved by Song and Tillmann. The main part of the proof is to show that $\Bϕ^{+} : \B B_{\infty}^{+} \rightarrow \B Γ_{\infty}^{+}$ induced from $ϕ$ 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 $ϕ: B_{n} \hookrightarrow Γ_{g,k}$. 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 $Φ: \mathcal{C} \rightarrow \mathcal{S}$ induced by $\Bϕ:\conf_n(D)\rightarrow\mathcal{M}_{g,k}$ preserves the actions of the framed little 2-disks operad.