Extendable periodic automorphisms of closed surfaces over the 3–sphere

A periodic automorphism of a surface † is said to be extendable over S³ if it extends to a periodic automorphism of the pair.†; S³/ for some possible embedding †,! S³. We classify and construct all extendable automorphisms of closed surfaces, with orientation-reversing cases included. Moreover, they can all be induced by automorphisms of S³ on Heegaard surfaces. As a byproduct, the embeddings of surfaces into lens spaces are discussed.