A survey of equivariant stable homotopy theory
type: Item
instance of: scholarly article
title: A survey of equivariant stable homotopy theory
zbMATH Open document ID: 0759.55001
DOI: 10.1016/0040-9383(92)90061-L
author: Gunnar Carlsson
published in: Topology
publication date: 1992-09-26T00:00:00Z
review text: Equivariant stable homotopy theory has evolved over the last 20-30 years into what seems nowadays to be a highly technical subject, based on an elaborate and complicated structure of foundational material. In view of this, the paper under review must be considered a highly welcome addition to the literature. To quote the author: ``It is the aim of this paper to give the reader a feeling for some of the issues that come up as one introduces \(G\)-actions into stable homotopy theory.'' This may be considered a slight understatement: in fact, the paper presents the main ideas and most important concepts of the field; it does this in a very clear and concise way, stressing the intuitive concepts and restricting to simple examples in cases where general statements would be burdened with technical and at first glance uncomprehensible assumptions. It thus can be recommended to anybody reasonably familiar with ordinary stable homotopy theory who wants to learn something about the modern developments in the equivariant theory. In accordance with the above-mentioned aim, the paper only considers the equivariant theory for finite groups \(G\). After some historical remarks in the introduction, the first section presents the basics of unstable equivariant homotopy theory. Since this section consists mostly of a straightforward generalization of the non- equivariant theory, it is rather brief, giving just the basic definitions and most important results. (An unexperienced reader should perhaps be warned that the author's notation \(X_ G\) for the usual orbit space \(X/G\) has a different meaning in most of the equivariant literature). The proper subject is introduced in section 2, which essentially treats the suspension category in the equivariant setting. According to the non- equivariant development, three lines of thought are followed: the definition of stable maps as maps to \(Q(X)=\varinjlim\Omega^ n\Sigma^ nX\), the construction of infinite loop spaces from symmetric monoi…
reviewed by: Erich Ossa
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P91
zbMATH DE Number: 54629
zbMATH Keywords: equivariant stable homotopy theory
zbMATH Keywords: suspension category
zbMATH Keywords: infinite loop spaces
zbMATH Keywords: framed bordism
zbMATH Keywords: configuration spaces
zbMATH Keywords: Burnside ring
zbMATH Keywords: suspension theorem
zbMATH Keywords: \(S\)-duality
zbMATH Keywords: transfer
zbMATH Keywords: Snaith-splitting
zbMATH Keywords: equivariant spectra
zbMATH Keywords: cohomology theories
zbMATH Keywords: tom Dieck splitting
zbMATH Keywords: Segal conjecture
zbMATH Keywords: Bredon coefficient systems
zbMATH Keywords: Mackey functors
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 54629