Hecke functors and the equivariant Dold-Thom theorem
type: Item
instance of: scholarly article
title: Hecke functors and the equivariant Dold-Thom theorem
zbMATH Open document ID: 0553.57015
author: Goro Nishida
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Let G be a finite group and S be a finite G-set. \({\mathbb{Z}}[S]\) denotes the free abelian group generated by S with the obvious G-action. Let k be a commutative ring with unit. A Hecke functor over k is an additive contravariant functor \(H: L^ G\to k\)-Mod, where \(L^ G\) is the category whose objects are all finite G-sets and \(L^ G(S,T)=Hom_{{\mathbb{Z}}[G]}({\mathbb{Z}}[S],{\mathbb{Z}}[T])\), and k-Mod is the category of k-modules. The author shows that the equivariant homotopy category \(Ho(M_{k[G]})\) of topological k[G]-modules and continuous k[G]-homomorphisms is determined in terms of Hecke functors. Moreover he shows that if the ring k is a PID such that 1/\(| G| \in k\), then the Bredon homotopy groups give an equivalence between the category \(Ho(M_{k[G]})\) and the category of the graded Hecke functors. This extends the Dold-Thom theorem [\textit{A. Dold} and \textit{R. Thom}, Ann. Math., II. Ser. 67, 239-281 (1958; Zbl 0091.371)] to the equivariant case.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55P20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3881505
zbMATH Keywords: Mackey functors
zbMATH Keywords: Eilenberg-MacLane spaces
zbMATH Keywords: infinite loop spaces
zbMATH Keywords: equivariant homotopy category of topological modules
zbMATH Keywords: equivariant Dold- Thom theorem
zbMATH Keywords: Hecke functor
zbMATH Keywords: finite G-sets
zbMATH Keywords: Bredon homotopy groups
reviewed by: Katsuhiro Komiya
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181828
OpenAlex ID: W2073464744
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DOI: 10.2977/PRIMS/1195181828
description: scientific article; zbMATH DE number 3881505