Group actions on spheres with rank one isotropy

Abstract

Abstract. Let G be a rank two finite group, and let H denote the family of all rank one p-subgroups of G for which rank p (G) = 2. We show that a rank two finite group G which satisfies certain explicit group-theoretic conditions admits a finite G-CW-complex X ≃ S n with isotropy in H, whose fixed sets are homotopy spheres. Our construction provides an infinite family of new non-linear G-CW-complex examples.

Homotopy representations over the orbit category

“…. Since E K (−) is defined as induction Ind F ′ (−) for the functor F ′ : R[W G (K)] → RΓ G/N (see [10,9.30]), we have…”

“…The algebraic homotopy representation conditions are easy to check locally over R = Z (p) at each prime, and fit well with the local-to-global procedure for constructing chain complexes C over ZΓ G . In a sequel [9] to this paper, we apply Corollary B to construct infinitely many new examples with rank one isotropy, for certain interesting families of rank two groups.…”

“…Moreover, character theory reveals intricate relations between the dimensions of the fixed subspheres S(V ) H , for subgroups H ≤ G, and the structure of the isotopy subgroups {G x | x ∈ S(V )}. Our goal is to better understand the constraints on these basic invariants, in order to construct new smooth non-linear finite group actions on spheres (see [8], [9]).…”