Orientation preserving actions of finite abelian groups on spheres

Abstract

If G G is a finite Abelian group acting as a Z ( P ) {{\mathbf {Z}}_{(\mathcal {P})}} -homology n n -sphere X X (where P \mathcal {P} is the set of primes dividing | G | ) |G|) , then there is an integer valued function n ( , G ) n(,G) defined on the prime power subgroups H H of G G such that X H {X^H} has the Z ( p ) {{\mathbf {Z}}_{(p)}} -homology of a sphere S n ( H , G ) {S^{n(H,G)}} . We prove here that there exists a real representation R R of G G such that for any prime power subgroup H H of G , dim ⁡ ( S ( R H ) ) = n ( H , G ) G,\dim (S({R^H})) = n(H,G) where S ( R H ) S({R^H}) is the unit sphere of R H {R^H} , provided that n − n ( H , G ) n - n(H,G) is even whenever H H is a 2 2 -subgroup of G G .