Extendable periodic automorphisms of closed surfaces over the 3-sphere

A periodic automorphism of a surface E is said to be extendable over S3 3 if it extends to a periodic automorphism of the pair (E, S3) 3 ) for some possible embedding E ,! ! S3. 3 . 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 3 on Heegaard surfaces. As a byproduct, the embeddings of surfaces into lens spaces are discussed.