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DOI: 10.1142/S1793525322500108

# Extending periodic maps on surfaces over the 4-sphere

Dec 2022

Let $F_g$ be the closed orientable surface of genus $g$. We address theproblem 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 ofmaximum order in ${\mathcal{M}}(F_g)$. Results including: (1) For each $g$, $w_g$ is periodically extendable over $S^4$ for somenon-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 extendableover $S^4$ 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 .

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