Pillar switchings and acyclic embedding in mapping class group

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The braid group Bg is embedded in the ribbon braid group that is defined to be the mapping class group Γ₀,₍g₎,₁. By gluing two copies of surface S₀,g₊₂ along g+1 holes, we get surface Sg,₁. A pillar switching is a self-homeomorphism of Sg,₁ which switches two pillars of surfaces by 180∘ horizontal rotation. We analyze the actions of pillar switchings on π₁ Sg,₁ and then give concrete expressions of pillar switchings in terms of standard Dehn twists. The map ψ: Bg → Γg,₁ sending the generators of Bg to pillar switchings on Sg,₁ is defined by extending the embedding Bg Γ₀,₍g₊₁₎,₁. We show that this map is injective by analyzing the actions of pillar switchings on π₁ Sg,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.