A linearity theorem for group actions on spheres with applications to homotopy representations
type: Item
instance of: scholarly article
title: A linearity theorem for group actions on spheres with applications to homotopy representations
zbMATH Open document ID: 0676.57016
DOI: 10.1007/BF02564667
author: Stefan Bauer
published in: Commentarii Mathematici Helvetici
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140146
review text: Let G be a finite group and \(T\subset {\mathbb{Q}}\) denote a subring which is contained in the local ring \({\mathbb{Z}}_{(p)}\) for any prime divisor p of the order of G. The author of the paper shows that if X is a finitistic G-space which is a T-homology sphere, then there exists a virtual representation V-W in the ring R(G) such that for any prime power order subgroup H of G, dim \(X^ H=\dim V^ H-\dim W^ H\). Then he applies this to give a complete description of the Grothendieck group of dimension functions of homotopy representations for compact Lie groups G. The description is provided in terms of dimension functions of linear representations as well as in combinatorial terms. In special cases, related results were obtained by \textit{R. M. Dotzel} and \textit{G. C. Hamrick} [Invent. Math. 62, 437-442 (1980; Zbl 0453.57030)] and \textit{T. tom Dieck} [Invent. Math. 67, 231-252 (1982; Zbl 0507.57026)].
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 4107758
zbMATH Keywords: finitistic G-space
zbMATH Keywords: homology sphere
zbMATH Keywords: dimension functions of homotopy representations for compact Lie groups
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4107758