Fusion systems and group actions with abelian isotropy subgroups

Details Fusion systems and group actions with abelian isotropy subgroups Fusion systems and group actions with abelian isotropy subgroups

2011-04-14

arxiv.org/abs/1104.2696v1

Mathematics

Algebraic Topology

21 pages

Scientific paper

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. We also discuss the conditions on a fusion system which imply the existence of a characteristic biset similar to the characteristic bisets associated to the saturated fusion systems.

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