Extending periodic maps on surfaces over the 4-sphere (Q6584806)

Considering orientation-preserving maps of a closed orientable surface \(F_g\), the authors consider the problem of when a periodic diffeomorphism of \(F_g\) extends to the 4-sphere, for some embedding of the surface into \(S^4\), or, stronger, \textit{extends periodically} to a map of the same finite order of \(S^4\). In particular, the authors consider the case of periodic diffeomorphisms of maximal possible order \(4g+2\) of a surface of genus \(g \ge 2\) (``Wiman maps'') (the case of extensions to the 3-sphere \(S^3\) is considered in a paper by \textit{C. Wang} et al. [Groups Geom. Dyn. 9, No. 4, 1001--1045 (2015; Zbl 1339.57027)]). The main results include the following.\N\NFor each \(g\), a Wiman map of order \(4g+2\) is \textit{periodically extendable} over \(S^4\) for some non-smooth embedding of \(F_g\) into \(S^4\), and not periodically extendable over \(S^4\) for any smooth embedding of \(F_g\) into \(S^4\).\N\NFor each \(g\), a Wiman map is \textit{extendable} over \(S^4\) for some embedding of \(F_g\) if and only if \(g = 4k, \; 4k+3\).\N\NEach torsion element of order \(p\) in the mapping class group of \(F_g\) is extendable over \(S^4\) for some smooth embedding of \(F_g\) if either \(p = 3^m\) and \(g\) is even, or \(p = 5^m\) and \(g \ne 4k+2\), or \(p = 7^m\) (where \(7\) can be replaced by infinitely many other primes); moreover, the conditions on \(g\) cannot be removed.