Symmetries of homology complex projective planes
type: Item
instance of: scholarly article
title: Symmetries of homology complex projective planes
zbMATH Open document ID: 0703.57024
DOI: 10.1007/BF02570739
author: Dariusz M. Wilczyński
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174130
review text: We consider here the question of which finite and compact Lie groups occur as groups of locally smooth symmetries of topological four- manifolds having the same integral homology as \({\mathbb{C}}P^ 2\), the complex projective plane. Our main result is that all such groups have a faithful projective representation of complex dimension 3 and therefore they also act linearly on \({\mathbb{C}}P^ 2\). In the case of homologically trivial actions this result has been previously shown by \textit{D. M. Wilczyński} [Ph. D. Thesis Indiana Univ. (1987)] and \textit{I. Hambleton} and \textit{R. Lee} [J. Algebra 116, 227-242 (1988; Zbl 0659.20041)]. The general principle behind recent results concerning locally smooth actions on homology \({\mathbb{C}}P^ 2\)'s is that the rigid topology of these manifolds forces the action to have the same local invariants as those of some linear action on \({\mathbb{C}}P^ 2\). This means that both actions have the same fixed points (if any), singular sets, local representations at the fixed points, etc.
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 20C10
zbMATH DE Number: 4153499
zbMATH Keywords: Chern manifold
zbMATH Keywords: groups of locally smooth symmetries of topological four- manifolds
zbMATH Keywords: complex projective plane
zbMATH Keywords: faithful projective representation of complex dimension 3
zbMATH Keywords: homology \({\mathbb{C}}P^ 2\)
zbMATH Keywords: linear action on \({\mathbb{C}}P^ 2\)
zbMATH Keywords: fixed points
zbMATH Keywords: singular sets
zbMATH Keywords: local representations at the fixed points
reviewed by: Amir H. Assadi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4153499