Construction of one-fixed-point actions on spheres of nonsolvable groups II

Abstract

Let 𝐺 be a finite group. If 𝑛 ≤ 5 then any 𝑛-dimensional homotopy sphere never admits a smooth action of 𝐺 with exactly one fixed point. Let 𝐴_𝑛 and 𝑆_𝑛 denote the alternating group and the symmetric group on some 𝑛 letters. If 𝑛 ≥ 6 then the 𝑛-dimensional sphere possesses a smooth action of 𝐴₅ with exactly one fixed point. Let 𝑉 be an 𝑛-dimensional real 𝐺-representation with exactly one fixed point. It is interesting to ask whether there exists a smooth 𝐺-action with exactly one fixed point on the 𝑛-dimensional sphere such that the associated tangential 𝐺-representation is isomorphic to 𝑉. In this paper, we study this problem for nonsolvable groups 𝐺 and real 𝐺-representations 𝑉 satisfying certain hypotheses. Applying a theory developed in this paper, we can prove that the 𝑛-dimensional sphere has an effective smooth action of 𝑆₅ with exactly one fixed point if and only if 𝑛 = 6, 10, 11, 12, or 𝑛 ≥ 14 and that the 𝑛-dimensional sphere has an effective smooth action of 𝐴₅ × 𝑍 with exactly one fixed point if 𝑛 satisfies 𝑛 ≥ 6 and 𝑛 ≠ 9, where 𝑍 is a group of order 2.