具有交换稳定子群的融合系统与群作用
Fusion systems and group actions with abelian isotropy subgroups
摘要：
证明了：若有限群$G$光滑地作用在流形$M$上，使得所有稳定子群都是秩为$\leq k$的阿贝尔群，则对某些正整数$n_1,...n_k$，$G$自由光滑地作用在$M \times \bbS^{n_1} \times...\times \bbS^{n_k}$上.我们构建这些行动使用递归方法，在早期的文件中介绍，涉及抽象的融合系统有限群。作为这个方法的另一个应用，我们证明了每个有限可解群都自由光滑地作用在球面的某个乘积上，而这些乘积对同调具有平凡作用。
英文摘要：
We prove that if a finite group $G$ acts smoothly on a manifold $M$ so that all the isotropy subgroups are abelian groups with rank $\leq k$, then $G$ acts freely and smoothly on $M \times \bbS^{n_1} \times...\times \bbS^{n_k}$ for some positive integers $n_1,...n_k$. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres with trivial action on homology.
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