Extendable periodic automorphisms of closed surfaces over the 3-sphere (Q6632133)

A periodic automorphism of a surface \(\Sigma\) is said to be extendable over \(S^3\) if it extends to a periodic automorphism of the pair \((\Sigma, S^3)\) for some embedding \(\Sigma \hookrightarrow S^3\). In the present paper, the authors classify and construct all automorphisms of closed surfaces, including the orientation-reversing case, which are extendable over \(S^3\); it turns out that for all of them \(\Sigma\) can be chosen to be a Heegaard surface of \(S^3\).\N\NThe analogous problem for \(\mathbb R^3\) instead of \(S^3\) is classical and considered in a paper by \textit{R. A. Rüedy} [in: Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 409--418 (1974; Zbl 0298.30020)]. The case of \(S^3\) was considered in a recent preprint by \textit{Y. Ni} et al. [Commun. Anal. Geom. 32, No. 3, 889--921 (2024; Zbl 07956698)]. Extendibility to \(S^3\) of arbitrary finite group actions on surfaces was considered in a paper by \textit{C. Wang} et al. [Groups Geom. Dyn. 9, No. 4, 1001--1045 (2015; Zbl 1339.57027)]; in contrast to the case of cyclic actions, there are extendable finite group actions on the surfaces \(\Sigma_{21}\) and \(\Sigma_{481}\) whose extensions cannot be realized by a Heegaard splitting.