Equivariant stable homotopy theory. With contributions by J. E. McClure.

*(English)*Zbl 0611.55001
Lecture Notes in Mathematics, 1213. Berlin etc.: Springer-Verlag. IX, 538 p. DM 80.00 (1986).

The aim of this book is to lay the foundations of G-equivariant stable homotopy theory. Such theories have been in use in a more or less naive form for some time and with good success, but mostly they have been restricted to G-CW-complexes for finite groups G. The present construction of the theory uses May’s approach to spectra and works for compact Lie groups G. We briefly indicate the basic notions.

First one chooses a good category of spaces, namely the category of compactly generated weak Hausdorff spaces, which has the advantage of possessing also function spaces with the usual adjunction homeomorphisms besides the categorical limits and colimits. Now let G be a compact Lie group. A category of G-spectra is constructed for each ”G-universe” U and ”indexing set” A; here U is an orthogonal representation of G containing each finite-dimensional subrepresentation finitely often and containing a trivial representation, whereas A is a family of finite-dimensional subrepresentations of U which contains an ascending sequence exhausting U.

A G-prespectrum D indexed on A then consists of based G-spaces D(V) for \(V\in A\) and based G-maps \(\sigma\) : \(\Sigma\) \({}^ wD(V)\to D(V\perp W)\) for V, \(V\perp W\in A\); it is called a G-spectrum if the adjoints \(\sigma\) : D(V)\(\to \Omega^ wD(V\perp W)\) are homeomorphisms. Maps are defined by the requirement of strict commutativity with the structure maps \(\sigma\). - The essential feature is now the existence of a left adjoint L to the inclusion functor from G-spectra to G-prespectra. Its importance is highlighted in Appendix A by a pointset-topology- construction in addition to the categorical existence proof; its usefulness comes from the possibility of transforming constructions of prespectra to constructions on spectra by applying L.

The advantages (and disadvantages) of such an approach are as in the non- equivariant theory. (In the reviewer’s opinion the advantages are here clearly in favour of this theory, even without the fact that at present there is no satisfactory competing product in the equivariant case). First, the category of spectra has easily constructed limits and small function-spectra. Colimits and small smash-products are constructed by application of L. The disadvantages are hidden in the fact that colimits may look somewhat different from the constructions one is used to. In fact, a good deal of the foundational material in this book consists in the careful examination of pointset-topology-properties for the constructions made in connection with the functor L.

The role of the universe U and the indexing set A is more important than one might suspect at first sight, since some of the constructions on spectra one needs have a natural tendency of changing the universe. Chapter II shows that on the other hand the choice of U and A is not too important since there are adjoint functors with good properties relating different choices. This chapter contains also a discussion of F-isotropic spectra for families F of subgroups as well as several important constructions like smash-products, function-spectra and functors related to a change of groups. A particular point worth emphasizing might be the nice treatment of Wirthmüller- and Adams-isomorphisms relating orbit- and fixed point constructions in the stable category.

Apart from topics on the category of spectra mentioned above, chapter I contains a discussion of the stable category obtained by inverting weak equivalences in the homotopy category of spectra. The usual calculus-of- fractions problems are circumvented by the construction of an equivalent category of G-CW-spectra, which can be built from stable cells \(G/H\times S^ n\) by a sequence of cofibrations.

The treatment of these matters is, however, somewhat sketchy, with proofs being essentially left to the reader. (One wishes that on p. 30 the authors had been able not only to quote but also to prove their version of Brown’s representability theorem. More seriously, they probably want their spacewise G-CW-approximation functor (in 5.12 and 6.3) to be continuous, and there seems to be no such functor in the published references.)

Chapter I, II (together with appendix A, treating the functor L) comprise the basic foundational material on the category of G-spectra. We shall now mention the further contents of the book more briefly.

Chapters VI, VII, VIII have to be seen (and perhaps should also be read) in conjunction with the companion volume [R. R. Bruner, J. P. May, J. E. McClure and M. Steinberger, \(H_{\infty}\) ring spectra and their applications (Lect. Notes Math. 1176) (1986; Zbl 0585.55016)]. They contain the basic foundational material on extended powers, operad ring spectra and their homological properties needed for the applications there. These notions depend on the categorical input of chapters I, II and on the new construction of the twisted half-smash product which is made choice independent by the use of spaces of linear isometries.

Chapters III, IV, V are devoted to the redevelopment of more or less well-known topics in equivariant homotopy theory within the current framework. While not containing essentially new results, they can be recommended as particularly nice and thorough treatments of respecively equivariant duality transfer maps and the theory of the Burnside ring and splitting theorems.

Chapter IX plays an exceptional role by being largely non-equivariant. The essential content is a precise and detailed construction of the Thom spectrum for a spherical fibration together with an investigation about the existence of additional structures of this spectrum. Chapter X finally gives an exposition of the theory of equivariant Thom spectra of G-vectorbundles.

Summarizing one can say that the book contains all the basic material on equivariant stable homotopy theory including as a particular merit, of course, the first construction of a satisfactory category of G-spectra. Its emphasis is on the categorical viewpoint, using the appropriate language and theory of adjoint functors for many of the useful constructions in equivariant homotopy theory. It will almost certainly become the basic standard reference in the field.

Whereas the book’s intention is the foundation of equivariant stable homotopy, it seems not to adress itself to the novice, but rather to the expert reader. This impression is supported by the style of writing, at least in respect to the organization of the presented material. Of course it could be argued that questions of style should be regarded as a matter of taste. Nevertheless the reviewer holds that especially in the more foundational chapters I, II, VI, VII the exposition could have been much more systematic. Many forward references, delays of important proofs, explanations in terms of notions which are developed later or elsewhere, references to unpublished sources make for hard reading if one has not yet obtained an overview over the whole contents. But still, for the reasons indicated above, it is certainly worth the trouble to work one’s way through this book.

First one chooses a good category of spaces, namely the category of compactly generated weak Hausdorff spaces, which has the advantage of possessing also function spaces with the usual adjunction homeomorphisms besides the categorical limits and colimits. Now let G be a compact Lie group. A category of G-spectra is constructed for each ”G-universe” U and ”indexing set” A; here U is an orthogonal representation of G containing each finite-dimensional subrepresentation finitely often and containing a trivial representation, whereas A is a family of finite-dimensional subrepresentations of U which contains an ascending sequence exhausting U.

A G-prespectrum D indexed on A then consists of based G-spaces D(V) for \(V\in A\) and based G-maps \(\sigma\) : \(\Sigma\) \({}^ wD(V)\to D(V\perp W)\) for V, \(V\perp W\in A\); it is called a G-spectrum if the adjoints \(\sigma\) : D(V)\(\to \Omega^ wD(V\perp W)\) are homeomorphisms. Maps are defined by the requirement of strict commutativity with the structure maps \(\sigma\). - The essential feature is now the existence of a left adjoint L to the inclusion functor from G-spectra to G-prespectra. Its importance is highlighted in Appendix A by a pointset-topology- construction in addition to the categorical existence proof; its usefulness comes from the possibility of transforming constructions of prespectra to constructions on spectra by applying L.

The advantages (and disadvantages) of such an approach are as in the non- equivariant theory. (In the reviewer’s opinion the advantages are here clearly in favour of this theory, even without the fact that at present there is no satisfactory competing product in the equivariant case). First, the category of spectra has easily constructed limits and small function-spectra. Colimits and small smash-products are constructed by application of L. The disadvantages are hidden in the fact that colimits may look somewhat different from the constructions one is used to. In fact, a good deal of the foundational material in this book consists in the careful examination of pointset-topology-properties for the constructions made in connection with the functor L.

The role of the universe U and the indexing set A is more important than one might suspect at first sight, since some of the constructions on spectra one needs have a natural tendency of changing the universe. Chapter II shows that on the other hand the choice of U and A is not too important since there are adjoint functors with good properties relating different choices. This chapter contains also a discussion of F-isotropic spectra for families F of subgroups as well as several important constructions like smash-products, function-spectra and functors related to a change of groups. A particular point worth emphasizing might be the nice treatment of Wirthmüller- and Adams-isomorphisms relating orbit- and fixed point constructions in the stable category.

Apart from topics on the category of spectra mentioned above, chapter I contains a discussion of the stable category obtained by inverting weak equivalences in the homotopy category of spectra. The usual calculus-of- fractions problems are circumvented by the construction of an equivalent category of G-CW-spectra, which can be built from stable cells \(G/H\times S^ n\) by a sequence of cofibrations.

The treatment of these matters is, however, somewhat sketchy, with proofs being essentially left to the reader. (One wishes that on p. 30 the authors had been able not only to quote but also to prove their version of Brown’s representability theorem. More seriously, they probably want their spacewise G-CW-approximation functor (in 5.12 and 6.3) to be continuous, and there seems to be no such functor in the published references.)

Chapter I, II (together with appendix A, treating the functor L) comprise the basic foundational material on the category of G-spectra. We shall now mention the further contents of the book more briefly.

Chapters VI, VII, VIII have to be seen (and perhaps should also be read) in conjunction with the companion volume [R. R. Bruner, J. P. May, J. E. McClure and M. Steinberger, \(H_{\infty}\) ring spectra and their applications (Lect. Notes Math. 1176) (1986; Zbl 0585.55016)]. They contain the basic foundational material on extended powers, operad ring spectra and their homological properties needed for the applications there. These notions depend on the categorical input of chapters I, II and on the new construction of the twisted half-smash product which is made choice independent by the use of spaces of linear isometries.

Chapters III, IV, V are devoted to the redevelopment of more or less well-known topics in equivariant homotopy theory within the current framework. While not containing essentially new results, they can be recommended as particularly nice and thorough treatments of respecively equivariant duality transfer maps and the theory of the Burnside ring and splitting theorems.

Chapter IX plays an exceptional role by being largely non-equivariant. The essential content is a precise and detailed construction of the Thom spectrum for a spherical fibration together with an investigation about the existence of additional structures of this spectrum. Chapter X finally gives an exposition of the theory of equivariant Thom spectra of G-vectorbundles.

Summarizing one can say that the book contains all the basic material on equivariant stable homotopy theory including as a particular merit, of course, the first construction of a satisfactory category of G-spectra. Its emphasis is on the categorical viewpoint, using the appropriate language and theory of adjoint functors for many of the useful constructions in equivariant homotopy theory. It will almost certainly become the basic standard reference in the field.

Whereas the book’s intention is the foundation of equivariant stable homotopy, it seems not to adress itself to the novice, but rather to the expert reader. This impression is supported by the style of writing, at least in respect to the organization of the presented material. Of course it could be argued that questions of style should be regarded as a matter of taste. Nevertheless the reviewer holds that especially in the more foundational chapters I, II, VI, VII the exposition could have been much more systematic. Many forward references, delays of important proofs, explanations in terms of notions which are developed later or elsewhere, references to unpublished sources make for hard reading if one has not yet obtained an overview over the whole contents. But still, for the reasons indicated above, it is certainly worth the trouble to work one’s way through this book.

Reviewer: E.Ossa

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P91 | Equivariant homotopy theory in algebraic topology |

55P42 | Stable homotopy theory, spectra |

57S10 | Compact groups of homeomorphisms |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |