Extending periodic maps on surfaces over the 4-sphere

Let \(F_g\) be the closed orientable surface of genus \(g\). We address the problem to extend torsion elements of the mapping class group \({\mathcal{M}}(F_g)\) over the 4-sphere \(S^4\). Let \(w_g\) be a torsion element of maximum order in \({\mathcal{M}}(F_g)\). Results including: (1) For each \(g\), \(w_g\) is periodically extendable over \(S^4\) for some non-smooth embedding \(e: F_g\to S^4\), and not periodically extendable over \(S^4\) for any smooth embedding \(e: F_g\to S^4\). (2) For each \(g\), \(w_g\) is extendable over \(S^4\) for some smooth embedding \(e: F_g\to S^4\) if and only if \(g=4k, 4k+3\). (3) Each torsion element of order \(p\) in \({\mathcal{M}}(F_g)\) is extendable over \(S^4\) for some smooth embedding \(e: F_g\to S^4\) if either (i) \(p=3^m\) and \(g\) is even; or (ii) \(p=5^m\) and \(g\ne 4k+2\); or (iii) \(p=7^m\). Moreover the conditions on \(g\) in (i) and (ii) can not be removed .