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Unstable modules over Steenrod algebra and Sullivan’s fixed point set conjecture. (English) Zbl 0871.55001
Chicago Lectures in Mathematics. Chicago, IL: Univ. of Chicago Press. x, 229 p., $ 40.00/hbk (1994).
This book provides a comprehensive account of some very important work in algebraic topology during the decade 1982-1992, dealing with properties of unstable modules over the Steenrod algebra and their application, via various generalized forms of the Adams spectral sequence, to deep theorems regarding the classification of mappings and the cohomology of mapping spaces. The book starts at a level appropriate for a graduate student just beginning research in algebraic topology, and, providing complete proofs of almost all results, takes the reader to the level of current mainstream research.
The fixed point theorem in the title, conjectured by Sullivan around 1970 and proved independently by Carlsson, Lannes, and Miller in the mid 1980’s, deals with a finite CW-complex \(X\) with a \(\mathbb{Z}/p\)-action. Define the homotopy fixed point set \(X^{h\mathbb{Z}/p}\) to be the space \(\text{map}_{\mathbb{Z}/p} (E \mathbb{Z}/p,X)\) of equivariant maps from a free contractible \(\mathbb{Z}/p\)-space to \(X\). The theorem states that the map \(X^{\mathbb{Z}/p} \to X^{h\mathbb{Z}/p}\) from the actual fixed point set to the homotopy fixed point set is an equivalence after suitable completion. A corollary when the action is trivial is that the space of pointed maps \(\text{map}_*(B \mathbb{Z}/p,X)\) is contractible.
Another major classification result in topology which is proved in this book is the following result conjectured by Miller and proved by Lannes. It states that if \(Y\) is connected and nilpotent, with \(\pi_1Y\) finite and \(H^*Y\) of finite type, and \(V\) is an elementary abelian \(p\)-group, (as it will be throughout this review), then the obvious morphism \([BV,Y]\to \operatorname{Hom}_{\mathcal K} (H^*Y, H^*V)\) is a bijection. Here \({\mathcal K}\) is the category of unstable algebras over the Steenrod algebra.
These results, along with other more technical results of a similar nature, are proved in the last third of the book. The proofs involve (co)simplicial techniques, including Anré-Quillen cohomology and the Bousfield-Kan spectral sequence. Some details in the development of the latter are omitted.
Essential to the proofs of these topological results are deep theorems about the category \({\mathcal U}\) of unstable modules over the Steenrod algebra. A thorough presentation of these results and their proofs occupy the first two-thirds of the book. One major topic here is Lannes’ \(T\)-functor. This is a functor \(T_V\) from the category \({\mathcal U}\) to itself, and a deep theorem shows that it induces a functor from \({\mathcal K}\) to itself. This functor \(T_V\) is left adjoint to the functor \(H^*V \otimes\). One of the ways that it relates to the applications is that, under mild hypotheses, there is an isomorphism \(T_VH^*X \to H^*\text{map} (BV,F_{p^\infty}X)\). The principal properties of \(T_V\), such as exactness and commuting with \(\otimes\), are proved, and some examples of calculating it are given. One example is \(T_VH^*W \approx H^*W^{\hom (V, W)}\) if \(W\) is an elementary abelian \(p\)-group.
Another major topic is the classification of injective objects of \({\mathcal U}\). Among these are the so-called Brown-Gitler modules \(J(n)\), which have the property that \(\hom_{\mathcal U} (M,J(n)) \approx M^{n*}\). Also, \(H^*V \otimes J(n)\) is injective. It is proved that every injective object of \({\mathcal U}\) can be written uniquely as a direct sum of modules \(L\otimes J(n)\), where \(n\) ranges over the set of natural numbers and \(L\) over the set of indecomposable summands of all \(H^*V\). This motivates one to want to classify the modules \(L\). They can be put into a 1-1 correspondence with indecomposable projective \(F_p[GL(V)]\)-modules, and some results about the Poincaré series of modules \(L\) are obtained from this.
Another topic which impacts the applications deals with nilpotent modules. A module \(M\) is nilpotent if (for \(p=2)\) iteration of \(x\mathop{\vrule height5pt depth0pt \vrule height2.7pt depth-2.3pt width12pt}\text{Sq}^{|x|}x\) eventually gives 0 for all \(x\) in \(M\). The full subcategory consisting of nilpotent modules is denoted Nil. One major theorem proves that the quotient category \({\mathcal U}/ \text{Nil}\) is isomorphic to the category of analytic functors. This latter category is a certain subcategory of the category of functors from the category of finite-dimensional vector spaces to the category of all vector spaces. As a simple example, the object \(H^*W\) in \({\mathcal U}/ \text{Nil}\) corresponds to the analytic functor \(V\mapsto F_p^{\hom (V,W)}\). Filtrations of \({\mathcal U}\) associated to Nil are defined. One nice application is the following result: If \(X\) is simply connected with \(H^*X\) of finite type, then \(H^*X\) is in \(\overline{\text{Nil}}_k\) if and only if \(\text{map}_*(BV,X)\) is \((k-1)\)-connected for any \(V\).
This book is jam packed with results about unstable \(A\)-modules and results about mapping spaces. It is clearly and thoroughly written. It could be of great benefit to graduate students or researchers in algebraic topology.

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55S10 Steenrod algebra