The generalized Harer conjecture for the homology triviality

TL;DR

This paper proves a generalized version of the Harer conjecture, demonstrating homology triviality for arbitrary embeddings of braid groups into mapping class groups by analyzing operad actions.

Contribution

It extends the classical Harer conjecture to all embeddings, introducing the concept of regular embeddings and using operad action preservation for the proof.

Findings

Homology triviality holds for all embeddings of braid groups into mapping class groups.

Regular embeddings suffice to prove the generalized conjecture.

Operad actions are preserved under the induced maps.

Abstract

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.