The generalized Harer conjecture for the homology triviality

초록

The classical Harer conjecture states the stable homology triviality of the canonical embedding phi:B2g+2 ↪Gamma g$\varphi :B_{2g+2}\hookrightarrow {\Gamma }_g$, which was proved by Song and Tillmann. The main part of the proof is to show that B phi+:BB infinity+-> B Gamma infinity+$B\varphi {<}^{>}+:B{B}_{\infty }{<}^{>}+\to B{\Gamma }_{\infty }{<}^{>}+$, induced from phi$\varphi$ is a double-loop space map. In this paper, we give a proof of the generalized Harer conjecture concerning the homology triviality for every regular embedding phi:Bn ↪Gamma g,k$\varphi :B_n\hookrightarrow {\Gamma }_{g,k}$. The main strategy of the proof is to remove all the interchangeable subsurfaces from Sg,k$S_{g,k}$ and collapse the new boundary components. Then, we obtain (the union of) covering spaces over a disk with n$n$ marked points that we can analyze. The final goal is to show that the map phi:C -> S$\Phi :\mathcal{C}\to \mathcal{S}$ induced by B phi:Confn(D)-> Mg,k$B\varphi :{\mathrm{Conf}}_n(D)\to {\mathcal{M}}_{g,k}$ preserves the actions of the framed little 2-disks operad.

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