Geometry and Topology Seminar

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) For infinitely many primes, each periodic map of order $p$ on $F_g$ is extendable over $S^4$ for some smooth embedding $e: F_g\to S^4$.

This is a joint work with Zhongzi Wang.