The generalized Harer conjecture for the homology triviality

The classical Harer conjecture is about the stable homology triviality of the obvious embedding \(\phi : B_{2g+2} \hookrightarrow \Gamma_{g}\), which was proved by Song and Tillmann. The main part of the proof is to show that \(\B\phi^{+} : \B B_{\infty}^{+} \rightarrow \B \Gamma_{\infty}^{+}\) indu...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Chang, Wonjun, Kim, Byung Chun, Song, Yongjin
Format: Article
Language:English
Subjects:
Disks
Embedding
Exchanging
Homology
Online Access:Get full text
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Summary: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.
ISSN:2331-8422