Extendability over the $4$-sphere and invariant spin structures of surface automorphisms

Abstract: It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus. Then each automorphism of an oriented once punctured surface is extendable over $S^4$. For each $g\neq 4$, we construct a periodic map on $F_g$ that is not extendable over $S^4$, and we prove that every periodic map on $F_4$ is extendable over $S^4$, which answer a question in \cite{WsWz}. We illustrate for an automorphism $f$ of $F_g$, how to find its invariant spin structures, bounding or not; and once $f$ has a bounding invariant spin structure, how to construct an embedding $F_g\hookrightarrow S^4$ so that $f$ is extendable with respect to this embedding.