Linearity of homotopy representations. II
type: Item
instance of: scholarly article
title: Linearity of homotopy representations. II
zbMATH Open document ID: 0809.57023
DOI: 10.1007/BF02567702
author: Ikumitsu Nagasaki
published in: Manuscripta Mathematica
publication date: 1995-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/155932
review text: [For Part I see Osaka J. Math. 29, No. 3, 595-606 (1992; Zbl 0778.57022).] Let \(G\) be a finite group. A finite \(G\)-CW complex \(X\) is called a homotopy representation of \(G\) if, for any subgroup \(H\) of \(G\), the \(H\)- fixed point set \(X^ H\) is homotopy equivalent to a \((\dim X^ H)\)- dimensional sphere or it is empty. Furthermore, if \(X\) is \(G\)-homotopy equivalent to a finite \(G\)-CW complex, it is called finite, if \(X\) is \(G\)-homotopy equivalent to a linear \(G\)-sphere, it is called linear, and if \(X* S(V)\) is linear for some linear \(G\)-sphere \(S(V)\), it is called stably linear. One defines the dimension function \(\text{Dim} X\) of \(X\) from the set of subgroups of \(G\) to the integers by setting \(\text{Dim} X(H) = \dim X^ H + 1\). If \(X^ H\) is empty, one sets \(\dim X^ H = - 1\). One calls \(\text{Dim} X\) linear if \(\text{Dim} X = \text{Dim} S(V)\) for some representation \(V\) of \(G\). The author proves the following results: Theorem A. All homotopy representations of \(G\) with linear dimension functions are linear if and only if \(G\) is isomorphic to \(C_{p^ m}\) \((m \geq 0)\), \(C_ 6\), \(D_{2p^ m}\) \((m \geq 1)\), \(D_{12}\), \(A_ 4\), or \(S_ 4\) \((p\) prime), where \(C_ n\) denotes the cyclic group of order \(n\), \(D_{2n}\) denotes the dihedral group of order \(2n\), and \(S_ n\) (resp. \(A_ n)\) denotes the symmetric group (resp. alternating group) on \(n\) letters. Corollary B. All homotopy representations of \(G\) are linear if and only if \(G\) is the cyclic \(p\) group \(C_{p^ m}\) or the dihedral 2 group \(D_{2^ m}\). Theorem C. Let \(G\) be an abelian group. Then all linear homotopy representations of \(G\) with linear dimension functions are linear if and only if \(G\) is an abelian \(p\)-group or \((C_ 2)^ m \times (C_ 3)^ n\) \((m,n \geq 1)\).
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 572228
zbMATH Keywords: finite \(G\)-CW complex
zbMATH Keywords: fixed point set
zbMATH Keywords: \(G\)-homotopy equivalent to a linear \(G\)-sphere
zbMATH Keywords: finite group
zbMATH Keywords: homotopy representation
zbMATH Keywords: stably linear
zbMATH Keywords: dimension function
zbMATH Keywords: linear dimension functions
zbMATH Keywords: cyclic group
zbMATH Keywords: dihedral group
zbMATH Keywords: symmetric group
zbMATH Keywords: alternating group
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
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OpenAlex ID: W3014491292
description: scientific article; zbMATH DE number 572228