From braid groups to mapping class groups

Chen, Lei; Mukherjea, Aru

doi:10.1007/s00209-022-03167-5

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2024

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“…Those interchanged subparts may be points or subsurfaces, not necessarily connected. Although the classification of embeddings from braid groups to mapping class groups is still widely open (e.g., [5]), all known embeddings are of this kind. Note that a standard generator of braid group is, in principle, characterized as interchanging two points by a half twist.…”

Section: Regular Embeddingsmentioning

confidence: 99%

The generalized Harer conjecture for the homology triviality

Chang,

Kim,

Song

2024

Bulletin of London Math Soc

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The 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.

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Section: Regular Embeddingsmentioning

confidence: 99%