Abelian $p$-group actions on homology spheres

Abstract:

The Borel formula is extended to an identity covering actions of arbitrary Abelian $p$-groups. Specifically, suppose $G$ is an Abelian $p$-group which acts on a finite ${\text {CW}}$-complex $X$ which is a ${Z_p}$-homology $n$-sphere. Each ${X^H}$ must be a ${Z_p}$-homology $n(H)$-sphere and then \[ n - n(G) = \sum {(n(K)} - n(K/p))\] where the sum is over ${A_0} = \{ \left . K \right |G/K\;{\text {is}}\;{\text {cyclic}}\}$ and the group $K/p$ is defined by \[ K/p = \{ g \in \left . G \right |pg \in K\} .\] This result is an immediate corollary of Theorem 2, whose converse Theorem 1, is also proven. Thus actions of Abelian $p$-groups on homology spheres resemble linear representations.