Pillar switchings and acyclic embedding in mapping class group

TL;DR

This paper introduces pillar switchings in mapping class groups, proves their injectivity from braid groups, and shows they induce trivial homology maps in the stable range using categorical delooping.

Contribution

It defines pillar switchings, analyzes their actions, and establishes their injectivity and homological triviality via categorical methods.

Findings

Pillar switchings are explicitly expressed in terms of Dehn twists.

The map from braid groups to mapping class groups is injective.

The induced homology map is trivial in the stable range.

Abstract

The braid group $B_g$ is embedded in the ribbon braid group that is defined to be the mapping class group $\Gamma_{0,(g),1}$. By gluing two copies of surface $S_{0,g+2}$ along $g+1$ holes, we get surface $S_{g,1}$. A pillar switching is a self-homeomorphism of $S_{g,1}$ which switches two pillars of surfaces by $180{}^\circ$ horizontal rotation. We analyze the actions of pillar switchings on $\pi_1 S_{g,1}$ and then give concrete expressions of pillar switchings in terms of standard Dehn twists. The map $\psi: B_g \rightarrow \Gamma_{g,1}$ sending the generators of $B_g$ to pillar switchings on $S_{g,1}$ is defined by extending the embedding $B_g \hookrightarrow \Gamma_{0,(g+1),1}$. We show that this map is injective by analyzing the actions of pillar switchings on $\pi_1 S_{g,1}$. The second part of this paper is to prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.