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THE PILLAR SWITCHINGS OF MAPPING CLASS GROUPS OF SURFACES

CHAN-SEOK JEONG

Departments of Mathematics, Inha University, Incheon 402-751, Korea

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and

YONGJIN SONG

Departments of Mathematics, Inha University, Incheon 402-751, Korea

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https://doi.org/10.1142/S0129167X13501036Cited by:3 (Source: Crossref)

Abstract

By gluing two copies of surface S_0,g+2 along g + 1 holes, we get surface S_g,1. The pillar switching is a self-homeomorphism of S_g,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π₁S_g,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : B_g → Γ_g,1 sending the generators of B_g to the pillar switchings on S_g,1 is defined by extending the embedding B_g ↪ Γ_0,(g+1),1. We show that this map is injective by analyzing the actions of pillar switchings on π₁S_g,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.

Keywords:

AMSC: 57M50, 55R37, 55P48

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