Group actions on homology spheres (Q1097551)

Let \(\pi\) be a finite group and \(\Sigma\) r a closed simply connected manifold which is a \({\mathbb{Z}}/[\pi]\) homology sphere. Let r be odd and not 3. The authors then show that \(\pi\) acts freely on the r-sphere if and only if \(\pi\) acts freely on \(\Sigma\) r with the induced action in homology being trivial. It should be noted that this result is about topological or PL actions and is false in the smooth case.