Orientation preserving actions of finite abelian groups on spheres

Abstract:

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