Embedding periodic maps on surfaces into those on S ³

Abstract

Call a periodic map h on the closed orientable surface Σ _g extendable if h extends to a periodic map over the pair (S ³,Σ _g) for possible embeddings e: Σ _g → S ³. The authors determine the extendabilities for all periodical maps on Σ₂. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair (S ³,Σ _g). To do this the authors first list all periodic maps on Σ₂, and indeed the authors exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be interesting. A by-product is that for each even g, the maximum order periodic map on Σ _g is extendable, which contrasts sharply with the situation in the orientation preserving category.

Keywords

Symmetry of surface / Symmetry of 3-sphere / Extendable action

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