
By gluing two copies of surface S0,g+2 along g + 1 holes, we get surface Sg,1. The pillar switching is a self-homeomorphism of Sg,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π1Sg,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : Bg → Γg,1 sending the generators of Bg to the 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 π1Sg,1. We also prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping method. 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.
55P48, 55R37, 57M50, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology
55P48, 55R37, 57M50, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology
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