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The Adams Spectral Sequence for Topological Modular Forms

Robert R. Bruner and John Rognes
Publication Date: 
Number of Pages: 
Mathematical Surveys and Monographs
[Reviewed by
Dan Isaksen
, on
One of the central problems of stable homotopy theory is the computation of the stable homotopy groups of spheres. This is a very hard problem for which there is no complete solution in sight. One way to formulate the problem is to define a stable homotopy category and to construct the "sphere spectrum" in this category. The homotopy groups of the sphere spectrum are the desired stable homotopy groups of spheres. In the stable homotopy category, there is the "topological modular forms" spectrum, tmf. This spectrum is interesting from several different perspectives. Its relevance for the present discussion is that its homotopy groups are a computable approximation to the stable homotopy groups of spheres.
The goal of the book is to give a self-contained, detail-oriented account of the stable homotopy groups of tmf. These computations have appeared in various places previously, but those accounts have provided incomplete details at best. In addition to computing the homotopy groups of tmf, the book also explains how to use these computations to deduce information about the stable homotopy groups of spheres. The authors use this approach to describe the stable homotopy groups of spheres in a large range.
The Adams spectral sequence is the main tool for computing the homotopy groups of tmf. This spectral sequence starts from a complicated algebraic Ext object that is defined entirely in terms of homological algebra. Part I of the text is dedicated to describing this algebraic object. The next and hardest step is to compute differentials in the Adams spectral sequence. This is the subject of Part II. Finally, one must reconstruct homotopy groups from an associated graded object. This last step is explained in Part III. The book contains many very interesting charts that present the computations graphically. These charts provide a coherent visualization of the computations. Similarly, the many tables provide the key computational facts.
The reviewer foresees at least two potential audiences for this text. First, experienced stable homotopy theorists can use the text as a reference for the homotopy groups of the sphere spectrum, of tmf, and of related spectra. Second, readers can use the manuscript as a textbook for learning how to compute in stable homotopy theory. One can use the book as a guide to reconstructing the computations. A reader who carried out this reconstruction would learn a tremendous amount about computational stable homotopy theory.

Dan Isaksen is Professor of Mathematics at Wayne State University.