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Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture (Chicago Lectures in Mathematics)

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I was impressed by the clarity of the first chapter, which serves as a gentle introduction of definitions and notations regarding unstable modules and algebras over the Steenrod algebra. While the following chapters tend to be more technical, I found this book a pleasant read. I recommend it to anyone who is a beginner in the subject of unstable modules [as I certainly am].

Fixed points of mappings are of course of enormous interested in the fields of dynamical systems and differential equations. This book however is interested in fixed points from the standpoint of algebraic topology, namely with the study of the homotopy type of the fixed point set of a group action. This motivates the consideration of a homotopy fixed point set, and the author studies specifically the homotopy fixed point set of a finite group acting on a finite complex. From an equivariant point of view, a homotopy fixed point is a set of maps equivariant under the integers modulo 2 (Z/2) from the "antipodal" sphere (i.e the ordinary sphere provided with the antipodal action) to a finite Z/2-CW-complex. A homotopy fixed point is such a map, and ordinary fixed points determine homotopy fixed points. The Sullivan fixed point conjecture asserts that the mapping of ordinary fixed points to homotopy fixed points is a homotopy equivalence, and this conjecture is one of the main topics of the book. The resolution of this conjecture was accomplished by Haynes Miller, with other contributions made by Gunnar Carlsson and J. Lannes. As a point of historical fact, Miller first proved the Sullivan conjecture in the context of pointed maps from the classifying space of a finite group to a finite CW-complex. With the compact-open topology, he showed that this space of maps is weakly contractible. Another line of thought on this topic and considered in this book is that of the fixed point set of a G-space localized at a prime p. The question of whether this fixed point set is weakly homotopy equivalent to the homotopy fixed point set of G acting on the p-localization of a finite CW-complex was solved by Carlsson via a consideration of the Segal conjecture.

The author gives a nice overview of these results, and does so by first considering background material from the theory of unstable modules over the Steenrod algebra. The reader is expected to have a solid background in algebraic topology, particularly in the homotopy theory of CW-complexes, Eilenberg-Maclane spaces, Postnikov systems, the theory of spectral reduced and unreduced cohomology, cohomology operations, and K-theory. The Steenrod algebra has its origins in the consideration of stable Z/2 cohomology operations, where these operations can all be written in terms of Steenrod operations. Consideration of relations among the Steenrod squares result in a family of relations called the Adem relations. This construction can be generalized to a prime p by considering generators other than the Steenrod squares, and dividing out the Adem relations (these are more complex than for the case p = 2). The calculation of the cohomology of Eilenberg-Maclane spaces leads to a characterization of the Steenrod algebra as the algebra of all transformations of mod p cohomology that commute with suspension. Such transformations are called 'stable'.
The mod p cohomology of a space as a module over the Steenrod algebra is unstable, meaning that it is trivial in negative degrees. The author then characterizes the category of unstable modules over the Steenrod algebra (designated U by the author), and shows that is has enough projectives and that it is (locally) Noetherian. That this category has enough injectives is shown using Brown-Gitler technology. This involves the construction of the Brown-Gitler modules, which are related to the Milnor algebra (the dual of the Steenrod algebra, familiar from the elementary theory), and the Carlsson modules. The later are related to Carlsson's work on the Segal conjecture, and their description involves some interesting use of the combinatorics of binary trees. The Lannes functor is introduced as a generalization of this tensor product that still gives an injective category, and its properties are outlined in detail. Modular representation theory is used in the book to study indecomposable reduced U-injectives, and their graded vector space structure is studied using the familiar Poincare series. Then the quotient category of U by its subcategory of nilpotents is studied via a filtration on it, the quotient categories of this filtration being identified with the modular representations of the symmetric groups.
The last part of the book finally gets down to the Sullivan conjecture, beginning with a discussion of the Andre-Quillen cohomology of unstable algebras over the Steenrod algebra. All of the familiar tools from algebraic topology, such as Eilenberg-Moore spectral sequences and the Borel construction are used to prove Miller's version of the Sullivan conjecture and also a generalized version of it.