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

TL;DR

This paper investigates when surface automorphisms extend over the 4-sphere, showing all automorphisms have invariant spin structures, and establishing conditions for extendability, including stable and periodic cases, with explicit constructions.

Contribution

It proves all surface automorphisms possess invariant spin structures and introduces stable extendability results, also classifying extendability of periodic maps on surfaces.

Findings

Every automorphism of a surface has an invariant spin structure.

All automorphisms of a once punctured surface are extendable over S^4.

Periodic maps on F_4 are all extendable over S^4.

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.