Pillar switchings and acyclic embedding in mapping class group

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.