Extending periodic maps on surfaces over the 4-sphere

Abstract

Let [Formula: see text] be the closed orientable surface of genus [Formula: see text]. We address the problem to extend torsion elements of the mapping class group [Formula: see text] over the 4-sphere [Formula: see text]. Let [Formula: see text] be a torsion element of maximum order in [Formula: see text]. Results including: (1) For each [Formula: see text], [Formula: see text] is periodically extendable over [Formula: see text] for some non-smooth embedding [Formula: see text], and not periodically extendable over [Formula: see text] for any smooth embedding [Formula: see text]. (2) For each [Formula: see text], [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if and only if [Formula: see text]. (3) Each torsion element of order [Formula: see text] in [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if either (i) [Formula: see text] and [Formula: see text] is even; or (ii) [Formula: see text] and [Formula: see text]; or (iii) [Formula: see text]. Moreover, the conditions on [Formula: see text] in (i) and (ii) cannot be removed.