论文标题
在四个球体上延长表面上的周期性地图
Extending periodic maps on surfaces over the 4-sphere
论文作者
论文摘要
令$ f_g $为$ g $属的封闭方向表面。我们解决了将映射类组$ {\ mathcal {m}}（f_g）$扩展到4-Sphere $ s^4 $的问题。令$ w_g $为$ {\ mathcal {m}}}（f_g）$中最大顺序的扭转元素。结果包括： （1）对于每种$ g $，$ w_g $对于某些非平滑嵌入$ e：f_g \ to s^4 $的$ s^4 $，可定期扩展，并且对于任何平滑嵌入$ e：f_g e：f_g e：f_g e：f_g f \ s^4 $ noce n ocky n ocky n of s^4 $ a：f_g \ \ a $ a：f_g \ to $ a：f_g \ $ a：f_g \ $ a：f_g \ $。 （2）对于每种$ g $，$ w_g $在$ s^4 $上可扩展，对于某些平滑嵌入$ e：f_g \ to s^4 $，并且仅当$ g = 4k，4k+3 $时。 （3）$ {\ MATHCAL {m}}}（f_g）$ in $ p $的每个扭转元素可在$ s^4 $上扩展，对于某些平滑嵌入$ e：f_g \ e：f_g \ to s^4 $，如果 （i）$ p = 3^m $和$ g $均匀； 或（ii）$ p = 5^m $和$ g \ ne 4k+2 $; 或（iii）$ p = 7^m $。 此外，无法删除（i）和（ii）的$ g $上的条件。
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 .
