Local surgery obstructions and space forms
Let \(\pi ={\mathbb{Z}}/m\rtimes \sigma\) be a metacyclic group with m odd, \(\sigma ={\mathbb{Z}}/2^ k\) and ker(t: \(\sigma\) \(\to ({\mathbb{Z}}/m)^{\times})\neq 1\). Then \(\pi\) has a free linear representation V with dim V\(=2q\), where \(q=2^{\ell}=| Im t|\). Let \(N=S(V)/\pi\) have k-invariant \(g(N)\in H^{2q}(\pi,{\mathbb{Z}})\), then if \((r,| \pi |)=1\) the class rg(N) is the first k-invariant (determining the homotopy type) of a free simplicial action of \(\pi\) on \(S^{2q-1}\). We study when such actions can be smoothed. Theorem 1. Let \(2q=2^{\ell +1}\geq 6\) and \(r\equiv 1 (mod 4)\). Assume \(| \ker t| =2\) and -1\(\in Im t\). Then \(\pi\) acts freely on \(S^{2q-1}\) with k-invariant rg(N) if and only if \(r\in ({\mathbb{Z}}/m)^{\times 2^{\ell}}.\) Theorem 2. Let \(\pi =Q(4m)\) be a quaternionic group of the above type \((k=2,\ell =1)\). Then if \(\pi\) acts freely on \(S^{4q-1}\) for any \(q\geq 1\), the action is homotopically linear. From the first result many nonlinear homotopy types of smooth actions occur. Also the condition given is necessary even for actions on \(S^ 3\). The second is an essential step in studying the existence of free actions of general periodic groups on spheres. The analogous problem for semi-free actions of \(\pi\) on \({\mathbb{R}}^{2q}\) fixing only the origin is also studied. Here the linear models are the representation spaces V. Necessary and sufficient conditions are given for all the metacyclic groups above, and again many nonlinear homotopy types arise.
semi-free actions on \({\mathbb{R}}^{2q}\)
free actions of general periodic groups on spheres
scientific article; zbMATH DE number 3925813
scientific article