On finite groups acting on \(\mathbb Z_2\)-homology 3-spheres
The authors obtain information about orientation-preserving actions of finite groups on integral homology \(3\)-spheres and \(\mathbb{Z}/2\)-homology \(3\)-spheres. The latter is an intermediate case between rational homology \(3\)-spheres, for which any finite group acts on some such manifold, and integral homology \(3\)-spheres, on which an extremely restricted class can act. The first main result says that if \(G\) acts on a \(\mathbb{Z}/2\)-homology \(3\)-sphere, then \(G\) can be obtained as follows: let \(\text{O}(G)\) be the maximal normal subgroup of \(G\) of odd order, then either \(G\) is solvable, or \(G/\text{O}(G)\) has a normal subgroup, with solvable factor group, isomorphic to one of \(\text{PSL}(2,q)\), \(\text{PSL}(2,q)\times\mathbb{Z}/2\), \(\text{SL}(2,q)\times_{\mathbb{Z}/2} C\), \(\text{SL}(2,q)\times_{\mathbb{Z}/2}\text{SL}(2,q')\), or \(\widehat{A}_7\), where \(C\) is solvable with a unique involution, \(q\) and \(q'\) are odd powers greater than four, and \(\widehat{A}_7\) is the unique perfect extension of the alternating group \(A_7\). Specializing the arguments for the first main result gives the second: a nonsolvable group acting on an integral homology \(3\)-sphere must be either \(A_5\), \(A_5\times \mathbb{Z}/2\), \(A_5^*\times_{\mathbb{Z}/2} C\), or \(A_5^*\times_{\mathbb{Z}/2} A_5^*\), where \(C\) is as before. There are additional restrictions, for example in the latter case the groups \(C\) must act freely, so must have periodic cohomology of period~\(4\) and appear on J. Milnor's well-known list of candidates to act freely. In neither case is it known that all groups of these forms do act. The authors have the impression that every group \(\text{SL}(2,q)\) and \(\text{PSL}(2,q)\) acts on some \(\mathbb{Z}/2\)-homology \(3\)-sphere, and have produced numerous examples of such actions. The proof of these results proceeds by using topological information to obtain various group-theoretic restrictions on \(G\), and then utilizing an impressive command of the theory of finite groups to find all groups satisfying them. Some of the important elements of the paper are drawn from work of M. Reni on the homology \(3\)-sphere case [\textit{M. Reni}, J. Lond. Math. Soc., II. Ser. 63, No. 1, 226--246 (2001; Zbl 1012.57025)], and from previous work of the second author. A key ingredient is the Gorenstein-Harada list of simple groups having sectional \(2\)-rank at most four (every \(2\)-subgroup is generated by at most four elements) and more than one conjugacy class of involutions.
scientific article; zbMATH DE number 2131999
scientific article