The generalized Harer conjecture for the homology triviality

Abstract

AbstractThe classical Harer conjecture states the stable homology triviality of the canonical embedding , which was proved by Song and Tillmann. The main part of the proof is to show that , induced from 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 . The main strategy of the proof is to remove all the interchangeable subsurfaces from and collapse the new boundary components. Then, we obtain (the union of) covering spaces over a disk with marked points that we can analyze. The final goal is to show that the map induced by preserves the actions of the framed little 2‐disks operad.