REMARKS ON DIMENSION OF HOMOLOGY SPHERES WITH ODD NUMBERS OF FIXED POINTS OF FINITE GROUP ACTIONS

Abstract

For each positive integer m, an arbitrary finite non-solvable group acts smoothly on infinitely many standard spheres with exactly m fixed points. However, for a given finite non-solvable group G and a given positive integer m, all standard spheres do not admit smooth actions of G with exactly m fixed points. In this paper, for each of the alternating group A₆ on six letters, the symmetric group S₆ on six letters, the projective general linear group PGL(2, 9) of order 720, the Mathieu group M₁₀ of order 720, the automorphism group Aut(A₆) of A₆ and the special linear group SL(2, 9) of order 720, we will give the dimensions of homology spheres whose fixed point sets of smooth actions of the group do not consist of odd numbers of points.