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Foundations of Stable Homotopy Theory

David Barnes and Constanze Roitzheim
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics
[Reviewed by
Dan Isaksen
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The goal of this book is to introduce the key ideas of stable homotopy theory to graduate students who are not yet familiar with the subject. Stable homotopy theory is a lively topic of current research, not just for its own sake but also for its applications to geometric topology, algebraic geometry, K-theory, and related fields. This is a notoriously difficult topic, and many graduate students have found its inaccessibility to be a
stumbling block to their graduate careers. On the other hand, a student who has mastered the subject finds that it is a gateway to a wide variety of research specialties. 
The authors have made great efforts to ensure that the book is accessible to those who are not already experts in the area. The topics have been carefully chosen, and the exposition includes not just the technical details but also provides historical and motivational context for many of the important ideas. The authors have wisely not included every detail and instead rely on references to other well-written accounts where appropriate.  The reader should beware that the book is far from elementary. A two-semester graduate course in algebraic topology is a minimum prerequisite.
Stability phenomena are some of the central organizing principles of algebraic topology. These phenomena appear in two related forms in introductory algebraic topology. First, the Freudenthal suspension theorem says that for a fixed \( k \), the homotopy classes of maps \( S^{n+k} \rightarrow S^{n} \) depends only on \( k \) and not on \( n \) for \( n \) sufficiently large. We call this a “stable” phenomenon because the homotopy classes are independent of \( n \).
Second, the homology and cohomology groups of a topological space are isomorphic to the homology and cohomology groups of its suspension with a shift of degree. This is also a “stable” phenomenon because the homology and cohomology groups of the \( n \)th suspension of a space are independent of \( n \).
Stable homotopy theory is an attempt to organize these stability phenomena in a categorically coherent way. From the perspective of homology and cohomology, a space is indistinguishable from its suspension.  Therefore, one might try to construct a category in which a space is actually equivalent to its suspension. This train of thought leads eventually to the Spanier-Whitehead category, which was the historically first attempt
at stable homotopy theory.
Unfortunately, the Spanier-Whitehead category has some technical deficiencies that prevent certain types of constructions. A major theme of algebraic topology in the second half of the twentieth century was to find categories for stable homotopy theory that avoid these deficiencies. This goal was only finally reached in the 1990’s with the advent of \( S \)-modules and symmetric spectra. The most difficult problem turned out to be the construction of a well-behaved smash product. Such a smash product is essential for a proper understanding of associativity, commutativity, and other structure motivated by analogies to classical algebra.
The book narrates the brief history of the last few paragraphs in much more detail. The reader will gain an understanding of the significance of stable homotopy theory, a grasp of how the technical details are arranged, and familiarity with the key results that are of use to the practicing stable homotopy theorist.


Dan Isaksen is Professor of Mathematics at Wayne State University.