Embedding periodic maps on surfaces into those on S ³

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. 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 (S³,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.