Embedding periodic maps on surfaces into those on S³

2013/02/05 by Yu Guo, Chao Wang, Guo, Yu +5
Mathematics · #57M60 #57S17 #57S25 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:57M60 #msc:57S17 #msc:57S25

paper · pdf · doi:10.48550/arxiv.1302.0972

22 pages, 21 figures

arxiv created 2013/02/05 · arxiv updated 2013/02/06

Abstract

Call a periodic map h on the closed orientable surface Σ_g extendable if h extends to a periodic map over the pair (S³, Σ_g) for possible embeddings e: Σ_g→ S³. We determine the extendabilities for all periodical maps on Σ₂. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair (S³, Σ_g). To do this we first list all periodic maps on Σ₂, and indeed we exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be an interesting piece. A by-product is that for each even g, the maximum order periodic map on Σ_g is extendable, which contrasts sharply to the situation in orientation preserving category.

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