On finite simple groups acting on homology 3-spheres
The main theorem is: The only nonabelian finite simple group which acts faithfully by diffeomorphisms on a homology 3-sphere is the alternating or dodecahedral group \(\mathbf A_5 \cong \text{PSL}(2,5) \). \textit{M. Reni} [J. Lond. Math. Soc., II. Ser. 63, No. 1, 226-246 (2001; Zbl 1012.57025)] and \textit{B. Zimmermann} [Monatsh. Math. 135, No. 3, 253-258 (2002; Zbl 0994.57019)] showed that a finite group \(G\) of orientation-preserving diffeomorphisms of a homology 3-sphere is either solvable or it has a unique composition factor which is isomorphic to \(\text{PSL}(2,q)\) for \(q\) an odd prime. In this paper the author shows: For a prime \(p\geq 5\), the group \(G=\text{PSL}(2,p)\) acts as a group of diffeomorphisms of homology 3-spheres if and only if \(p=5\), i.e. \(G\) is isomorphic to the dodecahedral group \(\mathbf A_5\).
scientific article; zbMATH DE number 1879119
scientific article