On the topological classification of pseudofree group actions on 4-manifolds: II
type: Item
instance of: scholarly article
title: On the topological classification of pseudofree group actions on 4-manifolds: II
zbMATH Open document ID: 0864.57012
DOI: 10.1007/BF00536601
author: Stefan Bauer
author: Dariusz M. Wilczyński
published in: \(K\)-Theory
publication date: 1997-06-22T00:00:00Z
review text: For Part I see [the second named author, Math. Z. 217, 335-366 (1994; Zbl 0813.57030)]. Let \(G\) be a finite cyclic group which acts (orientation preserving) locally linearly and pseudofreely on a closed oriented simply connected 4-manifold \(M\) with a nonempty fixed point set. Let \(O(\lambda)\) be the quotient of \(\Aut (H_2(M), \lambda)\) modulo those automorphisms inducing the identity on the Tate cohomology group \(\widehat H^0 (G;H_2(M))\). Similarly defined are the groups \(O(\lambda_r)\) for \(\lambda_r= \lambda\oplus rH\), where \(H\) is the hyperbolic form on \(\mathbb{Z}[G]^2\) and \(O_{st}(\lambda)= O(\lambda_r)\) with \(r\gg 0\). Then it follows that \[ O(\lambda) \subseteq O(\lambda_1) \subseteq O_{st} (\lambda) \subseteq O_*(\lambda), \] where the stable \(O\)-group \(O_*(\lambda)\) is the group of determinant \(\pm 1\) isometries of the associated quadratic form on \(\widehat H^0 (G;H_2 (M))\). The authors prove the following stability result of the equivariant intersection form \(\lambda\) of the \(G\)-action on \(M\): \[ O(\lambda_1)= O_{st} (\lambda)= O_*(\lambda). \] Then they obtain a topological rigidity theorem which states that two locally linear, pseudofree actions on \(M\), with the equivariant intersection forms indefinite and of rank at least 3 at each irreducible character, are topologically conjugate by an orientation preserving homeomorphism if and only if their oriented local representations at the corresponding fixed points are linearly equivalent.
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 19G38
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R91
zbMATH DE Number: 929693
zbMATH Keywords: pseudo-free action
zbMATH Keywords: finite cyclic group
zbMATH Keywords: simply connected 4-manifold
zbMATH Keywords: Tate cohomology
zbMATH Keywords: equivariant intersection forms
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00536601
OpenAlex ID: W3158542024
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description: scientific article; zbMATH DE number 929693