Finite group actions on a homotopy complex projective spaces
type: Item
instance of: scholarly article
title: Finite group actions on a homotopy complex projective spaces
zbMATH Open document ID: 0632.57026
DOI: 10.1007/BF01160217
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173958
review text: A homotopy \(CP^ n\) is a 2n-dimensional, smooth, closed, manifold which is homotopy equivalent to complex projective space \(CP^ n\). For \(n\geq 3\), it is known that there are infinitely many differentiably distinct homotopy \(CP^ n\)'s. The main result of this paper is that if G is an odd order group, or if G is 2-Sylow subgroup which is cyclic or \(Z_ 2\oplus Z_ 2\), or if G is a finite product of \(Z_ 2\)'s, then G acts smoothly and effectively on infinitely many differentiably distinct homotopy \(CP^ n\)'s, for some n. This is in contrast to the rigidity of Pontryagin classes conjectured for \(S^ 1\)-actions by \textit{T. Petrie} in his 1972 paper [Bull. Am. Math. Soc. 78, 105-153 (1972; Zbl 0247.57010)]. The major tool involved in the proof is equivariant surgery theory. The normal maps on which surgery is performed are endowed with a G-equivariant, orientation reversing involution thereby ensuring that the relevant equivariant signatures vanish.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 4028441
zbMATH Keywords: infinitely many differentiably distinct homotopy \(CP^ n\prime s\)
zbMATH Keywords: equivariant surgery theory
zbMATH Keywords: equivariant signatures
zbMATH Keywords: smooth group action
zbMATH Keywords: finite group actions
author: Mark C. Hughes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4028441