An action of a finite group on an $n$-cell without stationary points

E. E. Floyd, R. W. Richardson
1959 Bulletin of the American Mathematical Society  
If G is a transformation group on a space X, then x(~X is a stationary point if gx = x for every g£G. It has been an open problem, proposed by Smith [5] and by Montgomery [l, Problem 39], to determine whether every compact Lie group acting on a cell or on Euclidean space has a stationary point. Smith [4; 5] has shown the answer to be in the affirmative in case G is a toral group or a finite group of prime power order. In this note we give a simplicial action of A^ the group of even permutations
more » ... on five letters, on an n-cell without stationary points. Greever [3] has recently shown that the only finite groups of order less than 60 which could possibly act simplicially on a cell without stationary points are a certain class of groups of order 36. We wish to thank P. E. Conner for his help and encouragement. 1. The coset space SO(3)/I. Let SO(3) denote the group of all proper rotations of Euclidean 3-space E z and let IQSO(3) be the group of rotational symmetries of the icosahedron. As a group, / is isomorphic to A5 (see [9, pp. 16-18]) and hence is simple. LEMMA 1. The coset space 50(3)// has the integral homology groups of the 3-sphere S z . PROOF. Let Q denote the algebra of quaternions and QiQQ the group of quaternions of norm one. Identify Q with I? 4 and Qi with S z . Let r: Qi->50(3) be the standard homomorphism, which is a two-toone covering map. Set 7 / =T~* 1 (J). Then r induces a homeomorphism
doi:10.1090/s0002-9904-1959-10282-2 fatcat:r3wf6y7rw5borlqya2tjqzrheq