AI 中文总结
该研究针对复约化群中$m$元交换元素组空间的整奇异上同调，证明其挠素数与外尔群阶数的整除关系，解决了Kishimoto和Takeda的猜想，还推导了紧李群与紧李代数的对应结论。
AI 中文摘要
设$G$为复约化群，$C_m(G)_1$为$G$中$m$元交换元素组空间的单位分支。我们证明，对所有$m\boldsymbol{\text{≥}}2$，$C_m(G)_1$的整奇异上同调仅在那些整除$G$的外尔群阶数的素数处有挠。我们由此推导出紧李群的类似结论，解决了Kishimoto与Takeda的一个猜想，并证明了紧李代数的对应结果。我们利用素数幂阶半单元素共轭作用的史密斯理论来证明这些结论。
英文摘要
Let $G$ be a complex reductive group and $C_m(G)_1$ the identity component of the space of $m$-tuples of commuting elements in $G$. We prove that for every $m\geq 2$, the integral singular cohomology of $C_m(G)_1$ has torsion precisely at those primes which divide the order of the Weyl group of $G$. We deduce the analogous statement for compact Lie groups, settling a conjecture of Kishimoto and Takeda, and prove the corresponding result for compact Lie algebras. We prove these results using Smith theory for the conjugation action by semisimple elements of prime power order.
Comments16 pages, comments welcome
