On the genus of representation spheres
type: Item
instance of: scholarly article
title: On the genus of representation spheres
zbMATH Open document ID: 0704.57024
DOI: 10.1007/BF02566595
published in: Commentarii Mathematici Helvetici
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140184
review text: For \(G=S^ 1\) and a \(G\)-representation \(V\), in order to prove the existence of critical points of \(G\)-invariant functionals from the representation sphere \(SV\) into the real numbers, \textit{V. Benci} [Commun. Pure Appl. Math. 34, 393-432 (1981; Zbl 0447.34040)] used the notion of geometrical index of the \(G\)-space in question. For a topological group \(G\) and a \(G\)-space \(X\), the author of the paper considers the following two versions (closely related to the index) of the \(G\)-genus of \(X\). The \(G\)-genus \(\gamma_ G(X)\) (resp. \({\tilde \gamma}{}_ G(X))\) is the minimal number \(n\) such that for some isotropy subgroups \(H_ 1,\dots,H_ n\) which occur in \(X\) (resp., some proper closed subgroups \(H_ 1,\dots,H_ n\) of \(G\)), there exists a continuous equivariant map from \(X\) into the join \(G/H_ 1*\dots*G/H_ n\) [cf. \textit{E. Fadell}, Pac. J. Math. 89, 33-42 (1980; Zbl 0464.55004)]. For a cyclic group \(G\) and a \(G\)-representation \(V\) without trivial summand, the main result of the paper (Theorem 1.2) describes lower bounds for \(\gamma_ G(SV)\); and when the order of \(G\) is a prime power, it also gives lower bounds for \({\tilde \gamma}{}_ G(SV)\). As a corollary, the author obtains lower bounds for the \(G\)-genus of lens spaces with free \(G\)-actions when \(G\) is the cyclic group of order 2 [cf. \textit{A. Pfister} and \textit{S. Stolz}, Comment. Math. Helv. 62, 286-291 (1987; Zbl 0634.10020)]. The proof of the main result is based on calculations in equivariant \(K\)-theory. The author provides up to date information about related results and he concludes the paper with some open problems and remarks about the equivariant Lusternik-Schnirelmann category \(cat_ G(X)\) related to the genus \(\gamma_ G(X)\) so that \(cat_ G(X)\geq \gamma_ G(X)\).
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4155475
zbMATH Keywords: cyclic group actions
zbMATH Keywords: geometrical index
zbMATH Keywords: \(G\)-space
zbMATH Keywords: \(G\)-genus
zbMATH Keywords: \(G\)-genus of lens spaces with free \(G\)-actions
zbMATH Keywords: equivariant Lusternik-Schnirelmann category
author: Thomas J. Bartsch
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4155475