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

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 A(5) 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 S-5 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 A(5 )X Z with exactly one fixed point if n satisfies n > 6 and n not equal 9, where Z is a group of order 2.