Local surgery obstructions and space forms
type: Item
instance of: scholarly article
title: Local surgery obstructions and space forms
zbMATH Open document ID: 0578.57011
DOI: 10.1007/BF01174330
author: Ian Hambleton
author: Ib Madsen
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173777
review text: 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.
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57R67
zbMATH DE Number: 3925813
zbMATH Keywords: local surgery obstructions
zbMATH Keywords: space forms
zbMATH Keywords: free simplicial action on
zbMATH Keywords: \(S^{2q-1}\)
zbMATH Keywords: semi-free actions on \({\mathbb{R}}^{2q}\)
zbMATH Keywords: smoothing actions
zbMATH Keywords: metacyclic group
zbMATH Keywords: k-invariant
zbMATH Keywords: quaternionic group
zbMATH Keywords: homotopically linear
zbMATH Keywords: free actions of general periodic groups on spheres
zbMATH Keywords: nonlinear homotopy types
MaRDI profile type: Publication
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OpenAlex ID: W1968739623
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description: scientific article; zbMATH DE number 3925813