Extending periodic maps on surfaces over the 4-sphere

TL;DR

This paper investigates the extendability of torsion elements of the mapping class group of surfaces over the 4-sphere, revealing conditions under which such extensions are possible for smooth and non-smooth embeddings.

Contribution

It characterizes when torsion elements of the mapping class group can be extended over the 4-sphere for various embeddings and torsion orders, including maximum order elements.

Findings

Maximum order torsion elements are extendable over $S^4$ with non-smooth embeddings.

Extendability over $S^4$ for smooth embeddings occurs only for specific genus values.

Extendability depends on the order of torsion elements and the genus, with precise conditions identified.

Abstract

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 .