A survey of equivariant stable homotopy theory
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 monoidal categories, and the relation to framed bordism through the Pontrjagin- Thom construction. All these have their equivariant counterparts which are explained in detail. Along the way the author also discusses configuration spaces and the approximation theorem. Next the stable stems and the Burnside ring are introduced. The list of results familiar from the non-equivariant theory is then completed with the suspension theorem, \(S\)-duality, the transfer and the Snaith-splitting of \(Q(QX)\). One difference in the equivariant situation is the need for equivariant suspensions corresponding to representations \(V\) of \(G\). Thus \(S^ VX\) stands for the smash-product of the one-point compactification of the representation space \(V\) with the based \(G\)-space \(X\), and \(\Omega^ VX\) is the space of pointed maps from \(S^ V\) to \(X\); both are acted on by \(G\). The end of section 2 contains a thorough discussion of the \(G\)- spaces \(Q^ GX=\varinjlim\Omega^ V\Sigma^ VX\), where \(V\) runs now through all the finite-dimensional representations of \(G\). The equivariant stable homotopy groups of \(X\) may be considered as the ordinary homotopy groups of the fixed point space \((Q^ GX)^ G\). tom Dieck's splitting theorem for \((Q^ GX)^ G\) in terms of Weyl groups of subgroups of \(G\) is presented and a proof is sketched for \(G=\mathbb{Z}/2\). To make the picture complete it is shown finally that \(Q^ GX\) takes \(G\)- cofibrations into \(G\)-fibrations up to homotopy. The remainder of the paper is concerned with the heart matter of the subject. Section 3 explains equivariant spectra and cohomology theories while section 4 discusses the most important examples of these. In presenting \(G\)-spectra the author again aims at an understanding of the basic ideas. Thus technical matters which would be required for building up a complete machinery of \(G\)-spectra are omitted. A spectrum with \(G\)- actions is defined as a sequence of \(G\)-spaces \(X_ i\) together with equivalences \(X_ i\simeq \Omega X_{i+1}\), where in the loop space \(G\) acts trivially on the loop coordinate. On the other hand a \(G\)-spectrum is introduced as a sequence of \(G\)-spaces \(X_ i\) together with equivalences \(X_ i\simeq \Omega^ VX_{i+1}\), where \(V\) is now the regular representation of \(G\). The author explains how to relate these notions to each other, and how to define fixed point spectra and orbit spectra (which are ordinary spectra without \(G\)-action). Then the \(RO(G)\)-graded homology theories corresponding to \(G\)-spectra are introduced. At the end of this section the theme of the tom Dieck splitting is taken up again, now as an analysis of the fixed point spectrum for an arbitrary \(G\)-spectrum \(X\). Here the famous cofibration sequence \(EG_ +\to S^ 0\to\widetilde{EG}\) comes into play. This sequence was important in the author's solution of the Segal conjecture, which enters now through the discussion of the homotopy fixed point spectrum. The last section surveys various important \(G\)-homology theories. It starts with the introduction of Bredon coefficient systems and of Mackey functors. The latter are (according to Lewis, May and McClure) the coefficient systems for fully equivariant theories. (The definition of Mackey functor in this survey is, however, not the standard one, and might in fact be different). The homology theories considered then in detail are Borel (co-) homology, equivariant \(K\)-theory, stable homotopy, and equivariant bordism (with an eye towards the chromatic filtration of stable homotopy theory). The paper ends with a short list of problems in equivariant stable homotopy theory.
scientific article; zbMATH DE number 54629